P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP

Selman, Alan L.

doi:10.1007/3-540-09510-1_44

Cited by 57 publications

(93 citation statements)

References 14 publications

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“…We also note that the theory of P-selective sets [46] is vaguely related to our notion of an efficient selector. A set S C E* is P-selective if, for every pair (x, y), there is a polynomial time algorithm A : £* x S* -+ E* such that (a) A(x, y) € {x, y} and (b) (x € S or y e L) implies A(x, y) E L. In our case we have one instance at a time and need only decide what algorithm to run, a potentially simpler choice.…”

Section: Discussionmentioning

confidence: 99%

Confronting hardness using a hybrid approach

Vassilevska¹,

Williams²,

Woo³

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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A hybrid algorithm is a collection of heuristics, paired with a polynomial time selector S that runs on the input to decide which heuristic should be executed to solve the problem. Hybrid algorithms are interesting in scenarios where the selector must decide between heuristics that are "good" with respect to different complexity measures. In this paper, we focus on hybrid algorithms with a "hardness-defying" property: for a problem II, there is a set of complexity measures {rrii} whereby II is known or conjectured to be hard (or unsolvable) for each ra^, but for each heuristic hi of the hybrid algorithm, one can give a complexity guarantee for hi on the instances of II that S selects for hi that is strictly better than rrii. For example, we show that for NP-hard problems such as MAX-E/c-LlN-p, LONGEST PATH and MINIMUM BANDWIDTH, a given instance can either be solved exactly in "sub-exponential" (2°^) time, or be approximated in polynomial time with an approximation ratio exceeding that of the known or conjectured inapproximability of the problem, assuming P 7^ NP. We also prove some inherent limitations to the design of hybrid algorithms that arise under the assumption that NP requires exponential time algorithms.

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Section: Discussionmentioning

confidence: 99%

Confronting hardness using a hybrid approach

Vassilevska¹,

Williams²,

Woo³

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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“…A language L is P-selective [9], if there exists a polynomial-time Turing machine M such that for any strings x and y, the following conditions hold:…”

Section: Preliminariesmentioning

confidence: 99%

Robustness of PSPACE-complete sets

Pavan

Wang

2007

Information Processing Letters

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“…That in turn itself implies that if any NP-complete set belongs to P-sel then the polynomial hierarchy collapses, but the level to which it collapses the polynomial hierarchy is weaker than that to which the hierarchy is collapsed by the following early result of Selman [Sel79].…”

Section: Question 4: Where Does P-sel Fall In the Extended Low Hierarmentioning

confidence: 99%

“…However, it is not hard to see that this yields the same class as the following definition. Definition 1.1 ( [Sel79]) A set L is P-selective exactly if there is a (total, deterministic) polynomial-time function f such that, for all x, y ∈ Σ * : 1. f (x, y) ∈ {x, y}, and 2. {x, y} ∩ L = ∅ =⇒ f (x, y) ∈ L. * Supported in part by grant NSF-CCF-0426761.…”

Section: Introductionmentioning

confidence: 99%

Open questions in the theory of semifeasible computation

Faliszewski¹,

Hemaspaandra²

2006

SIGACT News

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The study of semifeasible algorithms was initiated by Selman's work a quarter of century ago [Sel79,Sel81,Sel82]. Informally put, this research stream studies the power of those sets L for which there is a deterministic (or in some cases, the function may belong to one of various nondeterministic function classes) polynomial-time function f such that when at least one of x and y belongs to L, then f (x, y) ∈ L ∩ {x, y}. The intuition here is that it is saying: "Regarding membership in L, if you put a gun to my head and forced me to bet on one of x or y as belonging to L, my money would be on f (x, y)."In this article, we present a number of open problems from the theory of semifeasible algorithms. For each we present its background and review what partial results, if any, are known.

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P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP

Cited by 57 publications

References 14 publications

Confronting hardness using a hybrid approach

Confronting hardness using a hybrid approach

Robustness of PSPACE-complete sets

Open questions in the theory of semifeasible computation

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