A local search algorithm solving an NP-complete optimization problem can be viewed as a stochastic process moving in an 'energy landscape' towards eventually finding an optimal solution. For the random 3-satis...
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A local search algorithm solving an NP-complete optimization 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 currently unsatisfied clauses 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 a sufficiently far below the critical satisfiability threshold alpha(c) approximate to 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 simple Metropolis dynamics. We estimate the optimal value for the 'temperature' parameter. for this algorithm, such that its linear time regime extends as close to alpha(c) as possible. Similar parameter optimization 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 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.
The multi-index matching problem generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyse its properties when the costs of the d-uplets ...
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The multi-index matching problem generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyse its properties when the costs of the d-uplets are random. At low temperatures we find for d >= 3 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.
A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disor...
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A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to compute exponentially small probabilities (rate functions) over different classes of random graphs. It is illustrated with two combinatorial optimization problems, the vertex-cover and colouring problems, for which the presence of replica symmetry breaking phases is taken into account. Applications include the analysis of models on adaptive graph structures.
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