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In [?], Schöning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n) · (4/3) n = O(1.3334 n) when given a formula F on n variables. This was the up to now best running time known for an algorithm solving 3-SAT. In this paper, we describe an algorithm… (More)

We consider the satisfiability problem on Boolean formulas in conjunctive normal form. We show that a satisfying assignment of a formula can be found in polynomial time with a success probability of 2 −n(1−1/(1+log m)) , where n and m are the number of variables and the number of clauses of the formula, respectively. If the number of clauses of the formulas… (More)

It is shown that the assumption of NP having polynomial-size circuits implies (apart from a collapse of the polynomial-time hierarchy as shown by Karp and Lip-ton) that the classes AM and MA of Babai's Arthur-Merlin hierarchy coincide. This means that also a certain inner collapse of the remaining classes of the polynomial-time hierarchy occurs. It is well… (More)

We consider the resource bounded measure of polynomial-time learnable subclasses of polynomial-size circuits. We show that if EXP 6 = MA, then every PAC-learnable subclass of P=poly has EXP-measure zero. We introduce a nonuniformly computable variant of resource bounded measure and show that, for every fixed polynomial q, any polynomial-time learnable… (More)

We first give añ O(2 n/3) quantum algorithm for the 0-1 Knapsack problem with n variables. More generally, for 0-1 Integer Linear Programs with n variables and d inequalities we give añ O(2 n/3 n d) quantum algorithm. For d = o(n/ log n) this running time is bounded by˜O(2 n(1/3+ǫ)) for every ǫ > 0 and in particular it is better than the˜O(2 n/2) upper… (More)

For the worst-case complexity measure, if P = NP, then P = OptP, i.e., all NP optimization problems are polynomial-time solvable. On the other hand, it is not clear whether a similar relation holds when considering average-case complexity. W e investigate the relationship between the complexity of NP decision problems and that of NP optimization problems… (More)

In this paper, we discuss the complexity and properties of the sets which are computable in polynomial-time on average. This study is motivated by Levin's question of whether all sets in NP are solvable in polynomial-time on average for every reasonable (i.e., polynomial-time computable) distribution on the instances. Let PP-comp denote the class of all… (More)