1,634 research outputs found

    Hiding solutions in random satisfiability problems: A statistical mechanics approach

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    A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem (3SAT). The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms.Comment: 5 pages, 4 figures, revised version to app. in PR

    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

    Integrating Conflict Driven Clause Learning to Local Search

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    This article introduces SatHyS (SAT HYbrid Solver), a novel hybrid approach for propositional satisfiability. It combines local search and conflict driven clause learning (CDCL) scheme. Each time the local search part reaches a local minimum, the CDCL is launched. For SAT problems it behaves like a tabu list, whereas for UNSAT ones, the CDCL part tries to focus on minimum unsatisfiable sub-formula (MUS). Experimental results show good performances on many classes of SAT instances from the last SAT competitions

    Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

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    Stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems. We give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems. We find two different dynamical regimes, depending on the number of constraints per variable: For low constraintness, the problems are solved efficiently, i.e. in linear time. For higher constraintness, the solution times become exponential. We observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium. If the algorithm runs long enough, an exponentially rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003

    On Improving Local Search for Unsatisfiability

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    Stochastic local search (SLS) has been an active field of research in the last few years, with new techniques and procedures being developed at an astonishing rate. SLS has been traditionally associated with satisfiability solving, that is, finding a solution for a given problem instance, as its intrinsic nature does not address unsatisfiable problems. Unsatisfiable instances were therefore commonly solved using backtrack search solvers. For this reason, in the late 90s Selman, Kautz and McAllester proposed a challenge to use local search instead to prove unsatisfiability. More recently, two SLS solvers - Ranger and Gunsat - have been developed, which are able to prove unsatisfiability albeit being SLS solvers. In this paper, we first compare Ranger with Gunsat and then propose to improve Ranger performance using some of Gunsat's techniques, namely unit propagation look-ahead and extended resolution

    Focused Local Search for Random 3-Satisfiability

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    A local search algorithm solving an NP-complete optimisation problem can be viewed as a stochastic process moving in an 'energy landscape' towards eventually finding an optimal solution. For the random 3-satisfiability problem, the heuristic of focusing the local moves on the presently unsatisfiedclauses is known to be very effective: the time to solution has been observed to grow only linearly in the number of variables, for a given clauses-to-variables ratio α\alpha sufficiently far below the critical satisfiability threshold αc≈4.27\alpha_c \approx 4.27. We present numerical results on the behaviour of three focused local search algorithms for this problem, considering in particular the characteristics of a focused variant of the simple Metropolis dynamics. We estimate the optimal value for the ``temperature'' parameter η\eta for this algorithm, such that its linear-time regime extends as close to αc\alpha_c as possible. Similar parameter optimisation is performed also for the well-known WalkSAT algorithm and for the less studied, but very well performing Focused Record-to-Record Travel method. We observe that with an appropriate choice of parameters, the linear time regime for each of these algorithms seems to extend well into ratios α>4.2\alpha > 4.2 -- much further than has so far been generally assumed. We discuss the statistics of solution times for the algorithms, relate their performance to the process of ``whitening'', and present some conjectures on the shape of their computational phase diagrams.Comment: 20 pages, lots of figure
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