Kuznetsov trace formula: Difference between revisions

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In [[computer science]], a '''property testing''' algorithm for a [[decision problem]] is an [[algorithm]] whose [[query complexity]] to its input is much smaller than the instance size of the problem.  Typically property testing algorithms are used to decide if some mathematical object (such as a [[graph (mathematics)|graph]] or a [[boolean function]]) has a "global" property, or is "far" from having this property, using only a small number of "local" queries to the object.
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For example, the following [[promise problem]] admits an algorithm whose query complexity is independent of the instance size (for an arbitrary constant ε > 0):
 
:"Given a graph ''G'' on ''n'' vertices, decide if ''G'' is [[Bipartite graph|bipartite]], or ''G'' cannot be made bipartite even after removing an arbitrary subset of at most <math>\epsilon\tbinom n2</math> edges of ''G''."
 
Property testing algorithms are important in the theory of [[probabilistically checkable proofs]].
 
==Definition and variants==
 
Formally, a '''property testing algorithm''' with query complexity ''q''(''n'') and ''proximity parameter'' ε for a decision problem ''L''  is a [[randomized algorithm]] that, on input ''x'' (an instance of ''L'') makes at most ''q''(|''x''|) queries to ''x'' and behaves as follows:
* If ''x'' is in ''L'', the algorithm accepts ''x'' with probability at least ⅔.
* If ''x'' is ε-far from ''L'', the algorithm rejects ''x'' with probability at least ⅔.
 
Here, "''x'' is ε-far from ''L''" means that the Hamming distance between ''x'' and any string in ''L'' is at least ε|''x''|.
 
A property testing algorithm is said to have ''one-sided error'' if it satisfies the stronger condition that the accepting probability for instances ''x ∈ L'' is 1 instead of ⅔.
 
A property testing algorithm is said be ''non-adaptive'' if it performs all its queries before it "observes" any answers to previous queries. Such an algorithm can be viewed as operating in the following manner.  First the algorithm receives its input. Before looking at the input, using its internal randomness, the algorithm decides which symbols of the input are to be queried. Next, the algorithm observes these symbols.  Finally, without making any additional queries (but possibly using its randomness), the algorithm decides whether to accept or reject the input.
 
==Features and limitations==
 
The main efficiency parameter of a property testing algorithm is its query complexity, which is the maximum number of input symbols inspected over all inputs of a given length (and all random choices made by the algorithm). One is interested in designing algorithms whose query complexity is as small as possible. In many cases the running time of property testing algorithms is [[Time complexity#Sub-linear time|sublinear]] in the instance length.  Typically, the goal is first to make the query complexity as small as possible as a function of the instance size ''n'', and then study the dependency on the proximity parameter ε.
 
Unlike other complexity-theoretic settings, the asymptotic query complexity of property testing algorithms is affected dramatically by the representation of instances.  For example, when ε = 0.01, the problem of testing bipartiteness of ''dense graphs'' (which are represented by their adjacency matrix) admits an algorithm of constant query complexity. In contrast, sparse graphs on ''n'' vertices (which are represented by their adjacency list) require property testing algorithms of query complexity <math>\Omega(\sqrt{n})</math>.
 
The query complexity of property testing algorithms grows as the proximity parameter ε becomes smaller for all non-trivial properties. This dependence on ε is necessary as a change of fewer than ε symbols in the input cannot be detected with constant probability using fewer than O(1/ε) queries. Many interesting properties of dense graphs can be tested using query complexity that depends only on ε and not on the graph size ''n''. However, the query complexity can grow enormously fast as a function of ε.  For example, for a long time the best known algorithm for testing if a graph does not [[Triangle-free graph|contain any triangle]] had a query complexity which is a [[Tetration|tower function]] of ''poly''(1/ε), and only in 2010 this has been improved to a tower function of ''log''(1/ε). One of the reasons for this enormous growth in bounds is that many of the positive results for property testing of graphs are established using the [[Szemerédi regularity lemma]], which also has tower-type bounds in its conclusions.
 
==References==
* [[Dana Ron]]. Property Testing. In Handbook of Randomization, 2000. [http://citeseer.ist.psu.edu/ron00property.html]
* Ronitt Rubinfeld. Sublinear-time Algorithms. In Proceedings of the 2006 International Congress of Mathematicians. [http://people.csail.mit.edu/ronitt/papers/icm.ps]
* [[Oded Goldreich]]. Combinatorial Property Testing (a survey). [http://www.wisdom.weizmann.ac.il/~oded/PS/testSU.ps][http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.39.6933&rep=rep1&type=pdf]
* Jacob Fox, A new proof of the graph removal lemma, to appear in Annals of Mathematics. [http://math.mit.edu/~fox/paper-removal.pdf]
 
[[Category:Approximation algorithms]]
[[Category:Randomized algorithms]]
[[Category:Theoretical computer science]]

Latest revision as of 22:38, 10 December 2014

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