Results 1 
4 of
4
Improving Exhaustive Search Implies Superpolynomial Lower Bounds
, 2009
"... The P vs NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been ..."
Abstract

Cited by 40 (7 self)
 Add to MetaCart
The P vs NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been derived from the assumption that an improvement exists. We show that there are natural NP and BPP problems for which minor algorithmic improvements over the trivial deterministic simulation already entail lower bounds such as NEXP ̸ ⊆ P/poly and LOGSPACE ̸ = NP. These results are especially interesting given that similar improvements have been found for many other hard problems. Optimistically, one might hope our results suggest a new path to lower bounds; pessimistically, they show that carrying out the seemingly modest program of finding slightly better algorithms for all search problems may be extremely difficult (if not impossible). We also prove unconditional superpolynomial timespace lower bounds for improving on exhaustive search: there is a problem verifiable with k(n) length witnesses in O(n a) time (for some a and some function k(n) ≤ n) that cannot be solved in k(n) c n a+o(1) time and k(n) c n o(1) space, for every c ≥ 1. While such problems can always be solved by exhaustive search in O(2 k(n) n a) time and O(k(n) + n a) space, we can prove a superpolynomial lower bound in the parameter k(n) when space usage is restricted.
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Worlds To Die For
, 1995
"... We survey the background and challenges of a number of open problems in the theory of relativization. Among the topics covered are pseudorandom generators, time hierarchies, the potential collapse of the polynomial hierarchy, and the existence of complete sets. Relativization (i.e., oracle) theory h ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We survey the background and challenges of a number of open problems in the theory of relativization. Among the topics covered are pseudorandom generators, time hierarchies, the potential collapse of the polynomial hierarchy, and the existence of complete sets. Relativization (i.e., oracle) theory has seen its share of ups and downs. Extensive surveys of current knowledge [Ver94] and debates as to relativization theory's merits [Har85,All90, HCC + 92,For94] can be found in the literature. However, in a nutshell, one could rather fairly say that as ups and downs go, relativization theory is on the mat. Still, that is not to say that relativization theory has no interesting open issues left with which to challenge theoretical computer scientists. It does, and here are a few such issues. Problem 1: Show that with probability one, the polynomial hierarchy is proper. The above statement is, to say the least, elliptic. However, the problem is wellknown in this formulation. The underlying...
PolynomialTime SemiRankable Sets
 Special Issue: Proceedings of the 8th International Conference on Computing and Information
, 1995
"... We study the polynomialtime semirankable sets (Psr), the ranking analog of the Pselective sets. We prove that Psr is a strict subset of the Pselective sets, and indeed that the two classes differ with respect to closure under complementation, closure under union with P sets, closure under join ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
We study the polynomialtime semirankable sets (Psr), the ranking analog of the Pselective sets. We prove that Psr is a strict subset of the Pselective sets, and indeed that the two classes differ with respect to closure under complementation, closure under union with P sets, closure under join with P sets, and closure under Pisomorphism. While P=poly is equal to the closure of Pselective sets under polynomialtime Turing reductions, we build a tally set that is not polynomialtime reducible to any Psr set. We also show that though Psr falls between the Prankable and the weaklyPrankable sets in its inclusiveness, it equals neither of these classes. Key words: semifeasible sets, Pselectivity, ranking, closure properties, NNT. 1 Introduction In the late 1970s, Selman [Sel79] defined the semifeasible (i.e., Pselective) sets, which are the polynomialtime analog of the Jockusch's [Joc68] semirecursive sets. Recently, there has been an intense renewal of interest in the P...