An ?(n 4/3) lower bound on the randomized complexity of graph properties

Hajnal, Peter I.

doi:10.1007/bf01206357

Cited by 26 publications

(13 citation statements)

References 10 publications

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“…Observe that all the variables have the same influence; thus, depending on whether the influence is large or small, one of the two bounds can be applied to yield the Ω(n 2/3 ) lower bound. This in particular reproduces the Ω(|V | 4/3 ) lower bound for all monotone graph properties in [5], which is O(log 1/3 (|V |)) shy of the current record [2].…”

Section: Introductionsupporting

confidence: 73%

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Jain¹,

Zhang²

2011

Theory of Comput.

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We give a simple proof, via query elimination, of a result due to O'Donnell, Saks, Schramm, and Servedio, which shows a lower bound on the zero-error expected query complexity of a function f in terms of the maximum influence of any variable of f . Our lower bound also applies to the two-sided error expected query complexity of f , and it allows an immediate extension which can be used to prove stronger lower bounds for some functions.

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Section: Introductionsupporting

confidence: 73%

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Jain¹,

Zhang²

2011

Theory of Comput.

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“…The query complexity of randomized decision procedures was conjectured by Yao to also be ⍀(N 2 ). Progress towards proving this conjecture was made in Yao [1987], King [1991], and Hajnal [1991] culminating in an ⍀(N 4/3 ) lower bound. This stands in striking contrast to the results mentioned above, by which some nontrivial monotone graph properties can be tested by examining a constant number of locations in the matrix.…”

Section: Introductionmentioning

confidence: 99%

Property testing and its connection to learning and approximation

Goldreich

Goldwasser

Ron

1998

J. ACM

906

850

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In this paper, we consider the question of determining whether a function f has property P or is ⑀-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a -Clique (clique of density with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

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“…It is discussed why all well-known randomized search heuristics are indeed black-box algorithms. Moreover, it turns out that blackbox algorithms can be described as randomized decision trees (a well-studied computational model in the context of boolean functions, see, e.g., Hajnal (1991), Heiman and Wigderson (1991), Heiman, Newman, and Wigderson (1993)).…”

Section: Introductionmentioning

confidence: 99%

Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization

2004

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Randomized search heuristics like local search, tabu search, simulated annealing or all kinds of evolutionary algorithms have many applications. However, for most problems the best worst-case expected run times are achieved by more problem-specific algorithms. This raises the question about the limits of general randomized search heuristics.Here a framework called black-box optimization is developed. The essential issue is that the problem but not the problem instance is known to the algorithm which can collect information about the instance only by asking for the value of points in the search space. All known randomized search heuristics fit into this scenario. Lower bounds on the black-box complexity of problems are derived without complexity theoretical assumptions and are compared to upper bounds in this scenario. * This work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the Collaborative Research Center "Computational Intelligence" (SFB 531).algorithms as well as for randomized algorithms (see, e.g., Cormen, Leiserson, and Rivest (1990) and Motwani and Raghavan (1995)). The criterion of the analysis is the asymptotic (w.r.t. the problem dimension), worst-case (w.r.t. the problem instance) expected (w.r.t. the random bits used by the algorithm) run time of the algorithm. Large lower bounds need some complexity theoretical assumption like NP = P or NP = RP. For almost all well-known optimization problems the best algorithms in this scenario are problem-specific algorithms which use the structure of the problem and compute properties of the specific problem instance.This implies that randomized search heuristics (local search, tabu search, simulated annealing, all kinds of evolutionary algorithms) are typically not considered in this context. They do not beat the highly specialized algorithms in their domain. Nevertheless, practitioners report surprisingly good results with these heuristics. Therefore, it makes sense to investigate these algorithms theoretically. There are theoretical results on local search (Papadimitriou, Schäffer, and Yannakakis (1990)). The analysis of the expected run time of the other search heuristics is difficult but there are some results (see, e.g., Glover and Laguna (1993) for tabu search, Kirkpatrick, Gelatt, and Vecchi (1983) and Sasaki and Hajek (1988) for simulated annealing, and Rabani, Rabinovich, and Sinclair (1998), Wegener (2001), Droste, Jansen, and Wegener (2002 and Giel and Wegener (2003) for evolutionary algorithms). Up to now, there is no "complexity theory for randomized search heuristics" which covers all randomized search heuristics and excludes highly specialized algorithms. Such an approach is presented in this paper.Our approach follows the tradition in complexity theory to describe and analyze restricted scenarios. There are well-established computation models like, e.g., circuits or branching programs (also called binary decision diagrams or BDDs) where one is not able to prove large lower bounds for explicitly defined problems. Therefore...

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An ?(n 4/3) lower bound on the randomized complexity of graph properties

Cited by 26 publications

References 10 publications

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Property testing and its connection to learning and approximation

Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization

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