13,438 research outputs found

    Single-Step Quantum Search Using Problem Structure

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    The structure of satisfiability problems is used to improve search algorithms for quantum computers and reduce their required coherence times by using only a single coherent evaluation of problem properties. The structure of random k-SAT allows determining the asymptotic average behavior of these algorithms, showing they improve on quantum algorithms, such as amplitude amplification, that ignore detailed problem structure but remain exponential for hard problem instances. Compared to good classical methods, the algorithm performs better, on average, for weakly and highly constrained problems but worse for hard cases. The analytic techniques introduced here also apply to other quantum algorithms, supplementing the limited evaluation possible with classical simulations and showing how quantum computing can use ensemble properties of NP search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with multiple steps (section 7). See also http://www.parc.xerox.com/dynamics/www/quantum.htm

    Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances

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    An analysis of the average-case complexity of solving random 3-Satisfiability (SAT) instances with backtrack algorithms is presented. We first interpret previous rigorous works in a unifying framework based on the statistical physics notions of dynamical trajectories, phase diagram and growth process. It is argued that, under the action of the Davis--Putnam--Loveland--Logemann (DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose characteristic parameters (ratio alpha of clauses per variable, fraction p of 3-clauses) can be followed during the operation, and define resolution trajectories. Depending on the location of trajectories in the phase diagram of the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are generated. Three regimes are identified, depending on the ratio alpha of the 3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost surely finds a solution in a time growing linearly with the number N of variables. Upper sat phase: for intermediate ratios, instances are almost surely satisfiable but finding a solution requires exponential time (2 ^ (N omega) with omega>0) with high probability. Unsat phase: for large ratios, there is almost always no solution and proofs of refutation are exponential. An analysis of the growth of the search tree in both upper sat and unsat regimes is presented, and allows us to estimate omega as a function of alpha. This analysis is based on an exact relationship between the average size of the search tree and the powers of the evolution operator encoding the elementary steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc

    Clustering of solutions in hard satisfiability problems

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    We study the structure of the solution space and behavior of local search methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the overlap measure of similarity between different solutions found on the same problem instance we show that the solution space is shrinking as a function of alpha. We consider chains of satisfiability problems, where clauses are added sequentially. In each such chain, the overlap distribution is first smooth, and then develops a tiered structure, indicating that the solutions are found in well separated clusters. On chains of not too large instances, all solutions are eventually observed to be in only one small cluster before vanishing. This condensation transition point is estimated to be alpha_c = 4.26. The transition approximately obeys finite-size scaling with an apparent critical exponent of about 1.7. We compare the solutions found by a local heuristic, ASAT, and the Survey Propagation algorithm up to alpha_c.Comment: 8 pages, 9 figure
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