Reducing the Number of Solutions of NP Functions

Hemaspaandra, Lane A.; Ogihara, Mitsunori; Wechsung, Gerd

doi:10.1006/jcss.2001.1815

Cited by 8 publications

(4 citation statements)

References 23 publications

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“…In addition to being of interest in their own right, they have recently had some surprising applications. For example, selectivity is a powerful tool in the study of search versus decision problems [HNOS96a], and nondeterministic generalizations of selectivity are the key tools used to show that even NP machines cannot uniquely refine satisfying assignments unless the polynomial hierarchy collapses [HNOS96b], that even weaker refinements are also precluded unless the polynomial hierarchy collapses [Ogi96,NRRS98], and that many cardinality types of nondeterministic function classes cannot collapse unless the polynomial hierarchy collapses [HOW02]. We call such a function f a P-selector function for B.…”

Section: Introductionmentioning

confidence: 99%

Algebraic Properties for Selector Functions

Hemaspaandra¹,

Hempel²,

Nickelsen³

2004

SIAM J. Comput.

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The nondeterministic advice complexity of the P-selective sets is known to be exactly linear.Regarding the deterministic advice complexity of the P-selective sets-i.e., the amount of KarpLipton advice needed for polynomial-time machines to recognize them in general-the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96].We prove that every associatively P-selective set is commutatively, associatively P-selective. Using this, we establish an algebraic sufficient condition for the P-selective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all P-selective sets are associatively P-selective then the deterministic advice complexity of the P-selective sets is linear. The weakest previously known sufficient condition was P = NP.We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.

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Section: Introductionmentioning

confidence: 99%

Algebraic Properties for Selector Functions

Hemaspaandra¹,

Hempel²,

Nickelsen³

2004

SIAM J. Comput.

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“…2 . These results were further improved by Hemaspaandra, Ogihara, and Wechsung [HOW02] who showed that Theorem 2.8 and Theorem 2.9 are a consequence of a more general lowness result. This was an improvement as if one believes that the polynomial hierarchy does not collapse, then Theorems 2.8 and 2.9 are of the form "false implies false."…”

Section: Question 3: How Far Does Np3v ⊆ C Np2v Collapse Ph?mentioning

confidence: 86%

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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“…Strengthening those collapses to a collapse to S NP∩coNP 2 remains open to this day and is a quite interesting challenge that has stymied many a graduate student. See also [HOW02] for some cases where refinements in fact are possible.…”

Section: Selectivity and Functions Team Up: Is Finding All Solutions ...mentioning

confidence: 99%

Beautiful Structures: An Appreciation of the Contributions of Alan Selman

Hemaspaandra

2014

Preprint

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Professor Alan Selman has been a giant in the field of computational complexity for the past forty years. This article is an appreciation, on the occasion of his retirement, of some of the most lovely concepts and results that Alan has contributed to the field. * This appreciation will appear, in slightly different form, in the Complexity Theory Column of the September 2014 issue of SIGACT News.† Supported in part by grant NSF-CCF-1101479.to the field are of such beauty and insight that they are as important to the field's future as they have been to its past and present. Any retirement is bittersweet, but Alan has mentioned that he will be keeping his hand in the field in retirement. That happy fact helped all of us at the celebration focus on the sweet side of the bittersweet event. Warm talks and memories were shared by everyone from the university's president, the department chair, and Alan's faculty colleagues all the way up to the people who are dearest of all to Alan-his postdocs and students.The warmth was no surprise. Alan is not just respected by but also is adored by those who have worked with him. Anyone who knows Alan knows why. Alan is truly kind, shockingly wise, and simply by his nature devoted to helping younger researchers better themselves and the field. But in fact, I think there is more to say-something far rarer than those all too rare characteristics. What one finds in Alan is a true belief in-an absolute, unshakable belief in-the importance of understanding of the foundations of the field. Now, one might think that Alan holds that belief as an article of faith. But my sense is that he holds the belief as an article of understanding. Like all the very, very best theoreticians, Alan has a terrific intuition about what is in the tapestry of coherent beauty that binds together the structure of computation. He doesn't see it all or even most of itno one ever has. But he knows it is there. And in these days when many nontheory people throw experiments and heuristics at hard problems, often without much of a framework for understanding behaviors or evaluating outcomes, not everyone can be said to even know that there is an organized, beautiful whole to be seen. Further, Alan has such a strong sense for what is part of the tapestry that-far more than most people-he has revealed the tapestry's parts and has guided his collaborators and students in learning the art of discovering pieces of the tapestry.And that brings us to the present article and its theme of the beauty of the structures and the structure that Alan has revealed-the notions, the directions, and the theorems. For all of us whose understanding isn't as deep as Alan's, the beauty of Alan's work has helped us to gain understanding, and to know that that tapestry really is out there, waiting to be increasingly discovered by the field, square inch by square inch, in a process that if it stretches beyond individual lifetimes nonetheless enriches the lifetimes of those involved in the pursuit of something truly important. To summarize Alan'...

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Reducing the Number of Solutions of NP Functions

Cited by 8 publications

References 23 publications

Algebraic Properties for Selector Functions

Algebraic Properties for Selector Functions

Open questions in the theory of semifeasible computation

Beautiful Structures: An Appreciation of the Contributions of Alan Selman

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