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Gauss–Bonnet theorem - Revision history
2024-03-28T10:36:32Z
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en>Yobot: /* Interpretation and significance */WP:CHECKWIKI error fixes using AWB (10093)
2014-05-05T11:50:36Z
<p><span dir="auto"><span class="autocomment">Interpretation and significance: </span><a href="/index.php?title=WP:CHECKWIKI&action=edit&redlink=1" class="new" title="WP:CHECKWIKI (page does not exist)">WP:CHECKWIKI</a> error fixes using <a href="/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a> (10093)</span></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Benton precisely what </del>you can <del style="font-weight: bold; text-decoration: none;">call me and Towards </del>the <del style="font-weight: bold; text-decoration: none;">gym comfortable </del>a <del style="font-weight: bold; text-decoration: none;">lot of use complete name</del>. <del style="font-weight: bold; text-decoration: none;">Louisiana is </del>in <del style="font-weight: bold; text-decoration: none;">addition </del>to <del style="font-weight: bold; text-decoration: none;">I love most these days I'm considering other programs</del>. <del style="font-weight: bold; text-decoration: none;">The favorite hobby </del>for <del style="font-weight: bold; text-decoration: none;">him </del>and <del style="font-weight: bold; text-decoration: none;">kids is jogging </del>it's <del style="font-weight: bold; text-decoration: none;">true he has time to adopt </del>on <del style="font-weight: bold; text-decoration: none;">new things</del>. <del style="font-weight: bold; text-decoration: none;">Production </del>and <del style="font-weight: bold; text-decoration: none;">planning is her day job now. 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Given below is one such test diet which allows you to consume 1400 calories in a day.<br><br>Please try other treatments plus food guides about eating proper to help you see the right system for you to get rid of weight quick. Here's the link to The Fat Loss Factor website if you would like to test it.</ins></div></td></tr>
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en>Yobot
https://en.formulasearchengine.com/index.php?title=Gauss%E2%80%93Bonnet_theorem&diff=224343&oldid=prev
en>Enyokoyama: /* Triangles */ avoid the ambiguity of "trigonometory."
2014-03-04T13:16:47Z
<p><span dir="auto"><span class="autocomment">Triangles: </span> avoid the ambiguity of "trigonometory."</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:16, 4 March 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{refimprove|date=August 2012}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Benton precisely what you </ins>can <ins style="font-weight: bold; text-decoration: none;">call me </ins>and <ins style="font-weight: bold; text-decoration: none;">Towards </ins>the <ins style="font-weight: bold; text-decoration: none;">gym comfortable </ins>a <ins style="font-weight: bold; text-decoration: none;">lot </ins>of <ins style="font-weight: bold; text-decoration: none;">use complete name</ins>. <ins style="font-weight: bold; text-decoration: none;">Louisiana </ins>is in <ins style="font-weight: bold; text-decoration: none;">addition </ins>to <ins style="font-weight: bold; text-decoration: none;">I love most </ins>these <ins style="font-weight: bold; text-decoration: none;">days I'm considering other programs</ins>. The <ins style="font-weight: bold; text-decoration: none;">favorite hobby </ins>for <ins style="font-weight: bold; text-decoration: none;">him </ins>and <ins style="font-weight: bold; text-decoration: none;">kids is jogging </ins>it<ins style="font-weight: bold; text-decoration: none;">'s true he has time </ins>to <ins style="font-weight: bold; text-decoration: none;">adopt on new things</ins>. <ins style="font-weight: bold; text-decoration: none;">Production </ins>and <ins style="font-weight: bold; text-decoration: none;">planning is her day job now</ins>. <ins style="font-weight: bold; text-decoration: none;">You </ins>can <ins style="font-weight: bold; text-decoration: none;">always find his website here</ins>: <ins style="font-weight: bold; text-decoration: none;">http</ins>:<ins style="font-weight: bold; text-decoration: none;">//www.yellowpages</ins>.<ins style="font-weight: bold; text-decoration: none;">com/nationwide/mip</ins>/<ins style="font-weight: bold; text-decoration: none;">paramount-solutions-inc</ins>-<ins style="font-weight: bold; text-decoration: none;">13528093?lid=13528093</ins><<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Also visit my webpage: </ins>[<ins style="font-weight: bold; text-decoration: none;">http</ins>://<ins style="font-weight: bold; text-decoration: none;">www.yellowpages</ins>.<ins style="font-weight: bold; text-decoration: none;">com</ins>/<ins style="font-weight: bold; text-decoration: none;">nationwide</ins>/<ins style="font-weight: bold; text-decoration: none;">mip</ins>/<ins style="font-weight: bold; text-decoration: none;">paramount</ins>-<ins style="font-weight: bold; text-decoration: none;">solutions</ins>-<ins style="font-weight: bold; text-decoration: none;">inc</ins>-<ins style="font-weight: bold; text-decoration: none;">13528093?lid</ins>=<ins style="font-weight: bold; text-decoration: none;">13528093 phone psychic readings</ins>]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The '''assignment problem''' is one of the fundamental [[combinatorial optimization]] problems in the branch of [[Optimization (mathematics)|optimization]] or [[operations research]] in [[mathematics]]. It consists of finding a maximum weight [[Matching (graph theory)|matching]] in a [[weighted graph|weighted]] [[bipartite graph]]. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In its most general form, the problem is as follows:</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:There are a number of ''agents'' and a number of ''tasks''. Any agent </del>can <del style="font-weight: bold; text-decoration: none;">be assigned to perform any task, incurring some ''cost'' that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the ''total cost'' of the assignment is minimized.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">If the numbers of agents and tasks are equal </del>and the <del style="font-weight: bold; text-decoration: none;">total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called the ''linear assignment problem''. Commonly, when speaking of the ''assignment problem'' without any additional qualification, then the ''linear assignment problem'' is meant.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Algorithms and generalizations ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The [[Hungarian algorithm]] is one of many [[algorithm]]s that have been devised that solve the linear assignment problem within time bounded by </del>a <del style="font-weight: bold; text-decoration: none;">polynomial expression of the number </del>of <del style="font-weight: bold; text-decoration: none;">agents</del>. <del style="font-weight: bold; text-decoration: none;"> </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The assignment problem is a special case of the [[transportation problem]], which </del>is <del style="font-weight: bold; text-decoration: none;">a special case of the [[minimum cost flow problem]], which </del>in <del style="font-weight: bold; text-decoration: none;">turn is a special case of a [[linear program]]. While it is possible </del>to <del style="font-weight: bold; text-decoration: none;">solve any of </del>these <del style="font-weight: bold; text-decoration: none;">problems using the [[simplex algorithm]], each specialization has more efficient algorithms designed to take advantage of its special structure. If the cost function involves quadratic inequalities it is called the [[quadratic assignment problem]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Example==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Suppose that a taxi firm has three taxis (the agents) available, and three customers (the tasks) wishing to be picked up as soon as possible</del>. The <del style="font-weight: bold; text-decoration: none;">firm prides itself on speedy pickups, so for each taxi the "cost" of picking up a particular customer will depend on the time taken </del>for <del style="font-weight: bold; text-decoration: none;">the taxi to reach the pickup point. The solution to the assignment problem will be whichever combination of taxis </del>and <del style="font-weight: bold; text-decoration: none;">customers results in the least total cost.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">However, the assignment problem can be made rather more flexible than </del>it <del style="font-weight: bold; text-decoration: none;">first appears. In the above example, suppose that there are four taxis available, but still only three customers. Then a fourth dummy task can be invented, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned </del>to <del style="font-weight: bold; text-decoration: none;">it</del>. <del style="font-weight: bold; text-decoration: none;">The assignment problem can then be solved in the usual way </del>and <del style="font-weight: bold; text-decoration: none;">still give the best solution to the problem</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Similar tricks </del>can <del style="font-weight: bold; text-decoration: none;">be played in order to allow more tasks than agents, tasks to which multiple agents must be assigned (for instance, a group of more customers than will fit in one taxi), or maximizing profit rather than minimizing cost.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Formal mathematical definition==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The formal definition of the '''assignment problem''' (or '''linear assignment problem''') is</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del style="font-weight: bold; text-decoration: none;">Given two sets, ''A'' and ''T'', of equal size, together with a [[weight function]] ''C'' </del>: <del style="font-weight: bold; text-decoration: none;">''A'' &times; ''T'' &rarr; '''[[real number|R]]'''</del>. <del style="font-weight: bold; text-decoration: none;">Find a [[bijection]] ''f'' : ''A'' &rarr; ''T'' such that the [[Loss function|cost function]]: </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">::<math>\sum_{a\in A}C(a,f(a))<</del>/<del style="font-weight: bold; text-decoration: none;">math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">is minimized.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Usually the weight function is viewed as a square real</del>-<del style="font-weight: bold; text-decoration: none;">valued [[matrix (mathematics)|matrix]] ''C'', so that the cost function is written down as:</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\sum_{a\in A}C_{a,f(a)}</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The problem can be expressed as a standard [</del>[<del style="font-weight: bold; text-decoration: none;">linear program]] with the objective function</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del style="font-weight: bold; text-decoration: none;"><math>\sum_{i\in A}\sum_{j\in T}C(i,j)x_{ij}<</del>/<del style="font-weight: bold; text-decoration: none;">math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">subject to the constraints</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>\sum_{j\in T}x_{ij}=1\text{ for }i\in A, \, <</del>/<del style="font-weight: bold; text-decoration: none;">math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>\sum_{i\in A}x_{ij}=1\text{ for }j\in T, \, </math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>x_{ij}\ge 0\text{ for }i,j\in A,T</del>. <del style="font-weight: bold; text-decoration: none;">\, </math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The variable <math>x_{ij}<</del>/<del style="font-weight: bold; text-decoration: none;">math> represents the assignment of agent <math>i<</del>/<del style="font-weight: bold; text-decoration: none;">math> to task <math>j<</del>/<del style="font-weight: bold; text-decoration: none;">math>, taking value 1 if the assignment is done and 0 otherwise. This formulation allows also fractional variable values, but there is always an optimal solution where the variables take integer values. This is because the constraint matrix is [[Unimodular matrix#Total unimodularity|totally unimodular]]. The first constraint requires that every agent is assigned to exactly one task, and the second constraint requires that every task is assigned exactly one agent.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==See also==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Auction algorithm]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Generalized assignment problem]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Linear bottleneck assignment problem]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[National Resident Matching Program]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Quadratic assignment problem]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Stable marriage problem]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Stable roommates problem]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Transportation theory (mathematics)|Monge-Kantorovich problem]], a more general formulation</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Weapon target assignment problem]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Further reading ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*{{cite book | last=Brualdi | first=Richard A. | title=Combinatorial matrix classes | series=Encyclopedia of Mathematics and Its Applications | volume=108 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0</del>-<del style="font-weight: bold; text-decoration: none;">521</del>-<del style="font-weight: bold; text-decoration: none;">86565</del>-<del style="font-weight: bold; text-decoration: none;">4 | zbl=1106.05001 }}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{cite book | authorlink </del>= <del style="font-weight: bold; text-decoration: none;">Rainer Burkard | first = Rainer | last = Burkard | coauthors = M. Dell'Amico, S. Martello | year = 2012 | title = Assignment Problems (Revised reprint) | publisher = SIAM | isbn = 978-1-61197-222-1 }}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Combinatorial optimization]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Matching]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Polynomial-time problems]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Linear programming]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[de:Zuordnungsproblem]</del>]</div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
en>Enyokoyama
https://en.formulasearchengine.com/index.php?title=Gauss%E2%80%93Bonnet_theorem&diff=2268&oldid=prev
en>Michael Hardy: /* Generalizations */
2013-08-31T17:11:31Z
<p><span dir="auto"><span class="autocomment">Generalizations</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:11, 31 August 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Did you think that it</del>'<del style="font-weight: bold; text-decoration: none;">s impossible to grow taller once you attain puberty. If you</del>'<del style="font-weight: bold; text-decoration: none;">re not taking </del>the <del style="font-weight: bold; text-decoration: none;">proper quantities and sorts </del>of <del style="font-weight: bold; text-decoration: none;">supplements</del>, <del style="font-weight: bold; text-decoration: none;">your body won't be growing taller. </del>There are a <del style="font-weight: bold; text-decoration: none;">range </del>of <del style="font-weight: bold; text-decoration: none;">natural things that you</del>'<del style="font-weight: bold; text-decoration: none;">ll </del>be <del style="font-weight: bold; text-decoration: none;">able </del>to <del style="font-weight: bold; text-decoration: none;">do so as to increase your height even after puberty</del>. <del style="font-weight: bold; text-decoration: none;">You just need </del>to <del style="font-weight: bold; text-decoration: none;">know the ways on how you can go about this </del>to <del style="font-weight: bold; text-decoration: none;">grow taller. Are you a busy person who doesn’t have time </del>to <del style="font-weight: bold; text-decoration: none;">complete </del>a <del style="font-weight: bold; text-decoration: none;">daily morning routine </del>and <del style="font-weight: bold; text-decoration: none;">exercise every evening </del>to <del style="font-weight: bold; text-decoration: none;">grow taller naturally. <br><br>To accommodate </del>the <del style="font-weight: bold; text-decoration: none;">multi-passenger crew </del>of the <del style="font-weight: bold; text-decoration: none;">Voskhod</del>, <del style="font-weight: bold; text-decoration: none;">potentially disastrous steps were made</del>. <del style="font-weight: bold; text-decoration: none;">Also</del>, <del style="font-weight: bold; text-decoration: none;">they will help in </del>the <del style="font-weight: bold; text-decoration: none;">proper functioning </del>of the <del style="font-weight: bold; text-decoration: none;">organs </del>which is <del style="font-weight: bold; text-decoration: none;">important for growing taller. Therefore</del>, <del style="font-weight: bold; text-decoration: none;">it cannot rest on it's own it need </del>a <del style="font-weight: bold; text-decoration: none;">firm place </del>of <del style="font-weight: bold; text-decoration: none;">gravitational pull while resting unlike </del>a <del style="font-weight: bold; text-decoration: none;">bed</del>. <del style="font-weight: bold; text-decoration: none;">You are also trying </del>to <del style="font-weight: bold; text-decoration: none;">avoid </del>any <del style="font-weight: bold; text-decoration: none;">spikes in your insulin levels while you are sleeping. Wire cage method: Get yourself a length </del>of <del style="font-weight: bold; text-decoration: none;">wire fencing. <br><br>Moreover</del>, <del style="font-weight: bold; text-decoration: none;">you will have </del>to <del style="font-weight: bold; text-decoration: none;">keep a good posture all day long if you want to grow taller</del>. If <del style="font-weight: bold; text-decoration: none;">this </del>is <del style="font-weight: bold; text-decoration: none;">not you</del>, <del style="font-weight: bold; text-decoration: none;">chances are you still could use some work</del>. <del style="font-weight: bold; text-decoration: none;">It should take approximately five </del>to <del style="font-weight: bold; text-decoration: none;">ten minutes </del>to <del style="font-weight: bold; text-decoration: none;">conduct both. There are a few reliable sources </del>of <del style="font-weight: bold; text-decoration: none;">information available on </del>the <del style="font-weight: bold; text-decoration: none;">web on grow taller exercises</del>, <del style="font-weight: bold; text-decoration: none;">but there are many, many </del>more <del style="font-weight: bold; text-decoration: none;">websites that do little, if anything, to fact-check the information they are getting from somewhere else before publishing </del>it. In <del style="font-weight: bold; text-decoration: none;">summary</del>, there are <del style="font-weight: bold; text-decoration: none;">numerous ways </del>of <del style="font-weight: bold; text-decoration: none;">growing taller even after </del>the <del style="font-weight: bold; text-decoration: none;">stipulated age</del>. <del style="font-weight: bold; text-decoration: none;"><br><br>These habits will create hindrance </del>in <del style="font-weight: bold; text-decoration: none;">your grow taller activities no matter whatever you do</del>. <del style="font-weight: bold; text-decoration: none;">Initial</del>, <del style="font-weight: bold; text-decoration: none;">you have got </del>to <del style="font-weight: bold; text-decoration: none;">concentrate on your eating routine and eating habits</del>. <del style="font-weight: bold; text-decoration: none;">This helps to tone </del>the <del style="font-weight: bold; text-decoration: none;">glutes </del>and <del style="font-weight: bold; text-decoration: none;">thighs</del>, <del style="font-weight: bold; text-decoration: none;">improving posture at </del>the <del style="font-weight: bold; text-decoration: none;">same time so you</del>. Usually, <del style="font-weight: bold; text-decoration: none;">after puberty </del>the <del style="font-weight: bold; text-decoration: none;">bones </del>in <del style="font-weight: bold; text-decoration: none;">legs get fused</del>, <del style="font-weight: bold; text-decoration: none;">but </del>the <del style="font-weight: bold; text-decoration: none;">bones </del>in <del style="font-weight: bold; text-decoration: none;">spine are not fused. Finally</del>, <del style="font-weight: bold; text-decoration: none;">if none of </del>the <del style="font-weight: bold; text-decoration: none;">above suits your needs</del>, <del style="font-weight: bold; text-decoration: none;">there are a few often overlooked tricks to employ that will give </del>the <del style="font-weight: bold; text-decoration: none;">appearance </del>of <del style="font-weight: bold; text-decoration: none;">added height. </del><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">My search ended when I found an effective way </del>to <del style="font-weight: bold; text-decoration: none;">make sure he can become as tall as he needs to be </del>and <del style="font-weight: bold; text-decoration: none;">without having to use any special medicines</del>. <del style="font-weight: bold; text-decoration: none;">Nevertheless</del>, <del style="font-weight: bold; text-decoration: none;">most of </del>the <del style="font-weight: bold; text-decoration: none;">details </del> is <del style="font-weight: bold; text-decoration: none;">focused on only </del>one <del style="font-weight: bold; text-decoration: none;">element. Since your body grows during sleep</del>, <del style="font-weight: bold; text-decoration: none;">it is important that you get at least 7 to 8 hours of uninterrupted sleep each night so as to gain those inches </del>and <del style="font-weight: bold; text-decoration: none;">wake up fresh in </del>the <del style="font-weight: bold; text-decoration: none;">morning. Choosing enough protein for </del>one<del style="font-weight: bold; text-decoration: none;">'s diet plays a large role in learning how to get taller</del>. <del style="font-weight: bold; text-decoration: none;">These tips will be </del>a <del style="font-weight: bold; text-decoration: none;">great start for you if you want to grow taller quickly</del>.<del style="font-weight: bold; text-decoration: none;"><br><br>In the event you loved this post </del>and <del style="font-weight: bold; text-decoration: none;">you would like to receive details regarding how to get taller fast </del>([<del style="font-weight: bold; text-decoration: none;">http</del>:<del style="font-weight: bold; text-decoration: none;">//www.estampas.info/ description here</del>]<del style="font-weight: bold; text-decoration: none;">) assure visit the web</del>-<del style="font-weight: bold; text-decoration: none;">page.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{refimprove|date=August 2012}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The '''assignment problem'</ins>'' <ins style="font-weight: bold; text-decoration: none;">is one of the fundamental [[combinatorial optimization]] problems in </ins>the <ins style="font-weight: bold; text-decoration: none;">branch of [[Optimization (mathematics)|optimization]] or [[operations research]] in [[mathematics]]. It consists </ins>of <ins style="font-weight: bold; text-decoration: none;">finding a maximum weight [[Matching (graph theory)|matching]] in a [[weighted graph|weighted]] [[bipartite graph]]. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In its most general form</ins>, <ins style="font-weight: bold; text-decoration: none;">the problem is as follows:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:</ins>There are a <ins style="font-weight: bold; text-decoration: none;">number of ''agents'' and a number </ins>of '<ins style="font-weight: bold; text-decoration: none;">'tasks''. Any agent can </ins>be <ins style="font-weight: bold; text-decoration: none;">assigned </ins>to <ins style="font-weight: bold; text-decoration: none;">perform any task, incurring some ''cost'' that may vary depending on the agent-task assignment</ins>. <ins style="font-weight: bold; text-decoration: none;">It is required </ins>to <ins style="font-weight: bold; text-decoration: none;">perform all tasks by assigning exactly one agent </ins>to <ins style="font-weight: bold; text-decoration: none;">each task and exactly one task </ins>to <ins style="font-weight: bold; text-decoration: none;">each agent in such </ins>a <ins style="font-weight: bold; text-decoration: none;">way that the ''total cost'' of the assignment is minimized.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If the numbers of agents and tasks are equal </ins>and <ins style="font-weight: bold; text-decoration: none;">the total cost of the assignment for all tasks is equal </ins>to the <ins style="font-weight: bold; text-decoration: none;">sum of the costs for each agent (or the sum </ins>of the <ins style="font-weight: bold; text-decoration: none;">costs for each task, which is the same thing in this case)</ins>, <ins style="font-weight: bold; text-decoration: none;">then the problem is called the ''linear assignment problem''</ins>. <ins style="font-weight: bold; text-decoration: none;">Commonly, when speaking of the ''assignment problem'' without any additional qualification</ins>, <ins style="font-weight: bold; text-decoration: none;">then the ''linear assignment problem'' is meant.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Algorithms and generalizations ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The [[Hungarian algorithm]] is one of many [[algorithm]]s that have been devised that solve the linear assignment problem within time bounded by a polynomial expression of </ins>the <ins style="font-weight: bold; text-decoration: none;">number of agents. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The assignment problem is a special case </ins>of the <ins style="font-weight: bold; text-decoration: none;">[[transportation problem]], </ins>which is <ins style="font-weight: bold; text-decoration: none;">a special case of the [[minimum cost flow problem]]</ins>, <ins style="font-weight: bold; text-decoration: none;">which in turn is </ins>a <ins style="font-weight: bold; text-decoration: none;">special case </ins>of a <ins style="font-weight: bold; text-decoration: none;">[[linear program]]</ins>. <ins style="font-weight: bold; text-decoration: none;"> While it is possible </ins>to <ins style="font-weight: bold; text-decoration: none;">solve </ins>any of <ins style="font-weight: bold; text-decoration: none;">these problems using the [[simplex algorithm]]</ins>, <ins style="font-weight: bold; text-decoration: none;">each specialization has more efficient algorithms designed </ins>to <ins style="font-weight: bold; text-decoration: none;">take advantage of its special structure</ins>. <ins style="font-weight: bold; text-decoration: none;"> </ins>If <ins style="font-weight: bold; text-decoration: none;">the cost function involves quadratic inequalities it </ins>is <ins style="font-weight: bold; text-decoration: none;">called the [[quadratic assignment problem]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Example==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Suppose that a taxi firm has three taxis (the agents) available</ins>, <ins style="font-weight: bold; text-decoration: none;">and three customers (the tasks) wishing to be picked up as soon as possible</ins>. <ins style="font-weight: bold; text-decoration: none;">The firm prides itself on speedy pickups, so for each taxi the "cost" of picking up a particular customer will depend on the time taken for the taxi </ins>to <ins style="font-weight: bold; text-decoration: none;">reach the pickup point. The solution </ins>to <ins style="font-weight: bold; text-decoration: none;">the assignment problem will be whichever combination </ins>of <ins style="font-weight: bold; text-decoration: none;">taxis and customers results in </ins>the <ins style="font-weight: bold; text-decoration: none;">least total cost.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">However</ins>, <ins style="font-weight: bold; text-decoration: none;">the assignment problem can be made rather </ins>more <ins style="font-weight: bold; text-decoration: none;">flexible than </ins>it <ins style="font-weight: bold; text-decoration: none;">first appears</ins>. In <ins style="font-weight: bold; text-decoration: none;">the above example</ins>, <ins style="font-weight: bold; text-decoration: none;">suppose that </ins>there are <ins style="font-weight: bold; text-decoration: none;">four taxis available, but still only three customers. Then a fourth dummy task can be invented, perhaps called "sitting still doing nothing", with a cost </ins>of <ins style="font-weight: bold; text-decoration: none;">0 for </ins>the <ins style="font-weight: bold; text-decoration: none;">taxi assigned to it</ins>. <ins style="font-weight: bold; text-decoration: none;">The assignment problem can then be solved </ins>in <ins style="font-weight: bold; text-decoration: none;">the usual way and still give the best solution to the problem</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Similar tricks can be played in order to allow more tasks than agents</ins>, <ins style="font-weight: bold; text-decoration: none;">tasks </ins>to <ins style="font-weight: bold; text-decoration: none;">which multiple agents must be assigned (for instance, a group of more customers than will fit in one taxi), or maximizing profit rather than minimizing cost</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Formal mathematical definition==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The formal definition of </ins>the <ins style="font-weight: bold; text-decoration: none;">'''assignment problem''' (or '''linear assignment problem''') is</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:Given two sets, ''A'' </ins>and <ins style="font-weight: bold; text-decoration: none;">''T''</ins>, <ins style="font-weight: bold; text-decoration: none;">of equal size, together with a [[weight function]] ''C'' : ''A'' &times; ''T'' &rarr; '''[[real number|R]]'''. Find a [[bijection]] ''f'' : ''A'' &rarr; ''T'' such that </ins>the <ins style="font-weight: bold; text-decoration: none;">[[Loss function|cost function]]: </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">::<math>\sum_{a\in A}C(a,f(a))</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">is minimized</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Usually <ins style="font-weight: bold; text-decoration: none;">the weight function is viewed as a square real-valued [[matrix (mathematics)|matrix]] ''C''</ins>, <ins style="font-weight: bold; text-decoration: none;">so that </ins>the <ins style="font-weight: bold; text-decoration: none;">cost function is written down as:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\sum_{a\</ins>in <ins style="font-weight: bold; text-decoration: none;">A}C_{a</ins>,<ins style="font-weight: bold; text-decoration: none;">f(a)}</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The problem can be expressed as a standard [[linear program]] with </ins>the <ins style="font-weight: bold; text-decoration: none;">objective function</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\sum_{i\in A}\sum_{j\</ins>in <ins style="font-weight: bold; text-decoration: none;">T}C(i</ins>,<ins style="font-weight: bold; text-decoration: none;">j)x_{ij}</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">subject to </ins>the <ins style="font-weight: bold; text-decoration: none;">constraints</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\sum_{j\in T}x_{ij}=1\text{ for }i\in A, \, </math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\sum_{i\in A}x_{ij}=1\text{ for }j\in T, \, </math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>x_{ij}\ge 0\text{ for }i,j\in A,T. \</ins>, <ins style="font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The variable <math>x_{ij}</math> represents </ins>the <ins style="font-weight: bold; text-decoration: none;">assignment </ins>of <ins style="font-weight: bold; text-decoration: none;">agent </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">i</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> to <ins style="font-weight: bold; text-decoration: none;">task <math>j</math>, taking value 1 if the assignment is done </ins>and <ins style="font-weight: bold; text-decoration: none;">0 otherwise</ins>. <ins style="font-weight: bold; text-decoration: none;"> This formulation allows also fractional variable values</ins>, <ins style="font-weight: bold; text-decoration: none;">but there is always an optimal solution where </ins>the <ins style="font-weight: bold; text-decoration: none;">variables take integer values. </ins> <ins style="font-weight: bold; text-decoration: none;">This is because the constraint matrix is [[Unimodular matrix#Total unimodularity|totally unimodular]]. </ins> <ins style="font-weight: bold; text-decoration: none;">The first constraint requires that every agent </ins>is <ins style="font-weight: bold; text-decoration: none;">assigned to exactly </ins>one <ins style="font-weight: bold; text-decoration: none;">task</ins>, and the <ins style="font-weight: bold; text-decoration: none;">second constraint requires that every task is assigned exactly </ins>one <ins style="font-weight: bold; text-decoration: none;">agent</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Auction algorithm]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Generalized assignment problem]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Linear bottleneck assignment problem]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[National Resident Matching Program]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Quadratic assignment problem]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Stable marriage problem]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Stable roommates problem]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Transportation theory (mathematics)|Monge-Kantorovich problem]], </ins>a <ins style="font-weight: bold; text-decoration: none;">more general formulation</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Weapon target assignment problem]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Further reading ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{cite book | last=Brualdi | first=Richard A</ins>. <ins style="font-weight: bold; text-decoration: none;">| title=Combinatorial matrix classes | series=Encyclopedia of Mathematics </ins>and <ins style="font-weight: bold; text-decoration: none;">Its Applications | volume=108 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86565-4 | zbl=1106.05001 }}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* {{cite book | authorlink = Rainer Burkard | first = Rainer | last = Burkard | coauthors = M. Dell'Amico, S. Martello | year = 2012 | title = Assignment Problems </ins>(<ins style="font-weight: bold; text-decoration: none;">Revised reprint) | publisher = SIAM | isbn = 978-1-61197-222-1 }}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Combinatorial optimization]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[</ins>[<ins style="font-weight: bold; text-decoration: none;">Category</ins>:<ins style="font-weight: bold; text-decoration: none;">Matching]</ins>]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Polynomial</ins>-<ins style="font-weight: bold; text-decoration: none;">time problems]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Linear programming]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[de:Zuordnungsproblem]]</ins></div></td></tr>
</table>
en>Michael Hardy
https://en.formulasearchengine.com/index.php?title=Gauss%E2%80%93Bonnet_theorem&diff=224342&oldid=prev
en>Mogism: Typo fixing and cleanup, typos fixed: repectively → respectively, dimentional → dimensional using AWB
2012-07-01T10:39:03Z
<p><a href="/index.php?title=WP:AWB/T&action=edit&redlink=1" class="new" title="WP:AWB/T (page does not exist)">Typo fixing</a> and cleanup, typos fixed: repectively → respectively, dimentional → dimensional using <a href="/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a></p>
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en>Mogism