Optimal heuristic algorithms for the image of an injective function

Hirsch, Edward; Itsykson, Dmitry; Nikolaenko, Valeria; Smal, Alexander

doi:10.1007/s10958-012-1102-y

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…A distributional problem is a pair of a language L and an ensemble of distributions D. A heuristic proof system [7] for a distributional problem (L, D) is an algorithm Π(x, w, δ; n) such that the following properties are satisfied:…”

Section: Heuristic Proofsmentioning

confidence: 99%

On Fast Heuristic Non-deterministic Algorithms and Short Heuristic Proofs

Itsykson

Sokolov

2014

Fundamenta Informaticae

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In this paper we study heuristic proof systems and heuristic non-deterministic algorithms. We give an example of a language Y and a polynomial-time samplable distribution D such that the distributional problem (Y, D) belongs to the complexity class HeurNP but Y / ∈ NP if NP = coNP, and (Y, D) / ∈ HeurBPP if (NP, PSamp) ⊆ HeurBPP.For a language L and a polynomial q we define the language pad q (L) composed of pairs (x, r) where x is an element of L and r is an arbitrary binary string of length at least q(|x|).is an ensemble of distributions on strings, let D × U q be a distribution on pairs (x, r), where x is distributed according to D n and r is uniformly distributed on strings of length q(n). We show that for every language L in AM there is a polynomial q such that for every distribution D concentrated on the complement of L the distributional problem (pad q (L), D × U q ) has a polynomially bounded heuristic proof system. Since graph non-isomorphism (GNI) is in AM, the above result is applicable to GNI.

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Section: Heuristic Proofsmentioning

confidence: 99%

On Fast Heuristic Non-deterministic Algorithms and Short Heuristic Proofs

Itsykson

Sokolov

2014

Fundamenta Informaticae

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“…In a parallel work [HINS11], we define simulations on the average in a different way: while Definition 4.2 is designed so that Levin-style Definition 4.1 of average-case polynomiality is closed under these simulations, the definitions of [HINS11] are intended to work with Impagliazzostyle average-case polynomiality [Imp95].…”

Section: Optimality On the Averagementioning

confidence: 99%

“…It was proved that for the language of Boolean tautologies, an optimal acceptor exists if and only if a p-optimal proof system exists; Messner generalized this result to paddable languages [Mes99]. While it is easy to see that Messner's proof goes for randomized acceptors as well, it does not apply to computations with advice (where optimal proof systems exist [CK07], but no optimal acceptors are known), heuristic computations (where optimal acceptors do exist [HINS11], but no optimal proof systems are known), or the restricted notion of optimality used in this paper. Recently, it was shown that the existence of optimal acceptors is equivalent for all co -NP-complete languages [CFM11].…”

Section: Introductionmentioning

confidence: 99%

On an optimal randomized acceptor for graph nonisomorphism

Hirsch

Itsykson

2012

Information Processing Letters

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An acceptor for a language L is an algorithm that accepts every input in L and does not stop on every input not in L. An acceptor Opt for a language L is called optimal if for every acceptor A for this language there exists polynomial p A such that for every x ∈ L, the running time time Opt (x) of Opt on x is bounded by p A (time A (x) + |x|) for every x ∈ L. (Note that the comparison of the running time is done pointwise, i.e., for every word of the language.)The existence of optimal acceptors is an important open question equivalent to the existence of p-optimal proof systems for many important languages [KP89, Sad99, Mes99]. Yet no nontrivial examples of languages in NP ∪ co -NP having optimal acceptors are known.In this short note we construct a randomized acceptor for graph nonisomorphism that is optimal up to permutations of the vertices of the input graphs, i.e., its running time on a pair of graphs (G 1 , G 2 ) is at most polynomially larger than the maximum (in fact, even the median) of the running time of any other acceptor taken over all permuted pairs (π 1 (G 1 ), π 2 (G 2 )). One possible motivation is the (pointwise) optimality in the class of acceptors based on graph invariants where the time needed to compute an invariant does not depend much on the representation of the input pair of nonisomorphic graphs.In fact, our acceptor remains optimal even if the running time is compared to the average-case running time over all permuted pairs. We introduce a natural notion of average-case optimality (not depending on the language of graph nonisomorphism) and show that our acceptor remains average-case optimal for any probability distribution on the inputs that respects permutations of vertices.

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“…Approximate algorithms are able to find good answers (near to optimal solutions) for NP-hard in a short time. They are divided into two groups of heuristic algorithms [8][9][10] and meta-heuristic algorithms [11]. Heuristic approaches create suitable and good solutions which normally are not the best solution.…”

Section: Introductionmentioning

confidence: 99%

Solving optimization problems using black hole algorithm

Farahmandian¹,

Hatamlou²

2015

JACST

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Various meta-heuristic optimization approaches have been recently created and applied in different areas. Many of these approaches are inspired by swarm behaviors in the nature. This paper studies the solving optimization problems using Black Hole Algorithm (BHA) which is a population-based algorithm. Since the performance of this algorithm was not tested in mathematical functions, we have studied this issue using some standard functions. The results of the BHA are compared with the results of GA and PSO algorithms which indicate that the performance of BHA is better than the other two mentioned algorithms.

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Optimal heuristic algorithms for the image of an injective function

Cited by 3 publications

References 8 publications

On Fast Heuristic Non-deterministic Algorithms and Short Heuristic Proofs

On Fast Heuristic Non-deterministic Algorithms and Short Heuristic Proofs

On an optimal randomized acceptor for graph nonisomorphism

Solving optimization problems using black hole algorithm

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