The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective

Stochastic local search (SLS) algorithms have shown effectiveness on satisfiable instances of the Boolean satisfiability (SAT) problem. However, their performance is still unsatisfactory on random k-SAT at the phase transition, which is of significance and is one of the empirically hardest distributions of SAT instances. In this paper, we propose a new heuristic called DCCA, which combines two configuration checking (CC) strategies with different definitions of configuration in a novel way. We use the DCCA heuristic to design an efficient SLS solver for SAT dubbed DCCASat. The experiments show that the DCCASat solver significantly outperforms a number of state-of-the-art solvers on extensive random k-SAT benchmarks at the phase transition. Moreover, DCCASat shows good performance on structured benchmarks, and a combination of DCCASat with a complete solver achieves state-of-the-art performance on structured benchmarks.

Double configuration checking in stochastic local search for satisfiability

Approximate message passing algorithm enjoyed considerable attention in the last decade. In this paper we introduce a variant of the AMP algorithm that takes into account glassy nature of the system under consideration. We coin this algorithm as the approximate survey propagation (ASP) and derive it for a class of low-rank matrix estimation problems. We derive the state evolution for the ASP algorithm and prove that it reproduces the one-step replica symmetry breaking (1RSB) fixed-point equations, well-known in physics of disordered systems. Our derivation thus gives a concrete algorithmic meaning to the 1RSB equations that is of independent interest. We characterize the performance of ASP in terms of convergence and mean-squared error as a function of the free Parisi parameter s. We conclude that when there is a model mismatch between the true generative model and the inference model, the performance of AMP rapidly degrades both in terms of MSE and of convergence, while ASP converges in a larger regime and can reach lower errors. Among other results, our analysis leads us to a striking hypothesis that whenever s (or other parameters) can be set in such a way that the Nishimori condition M = Q > 0 is restored, then the corresponding algorithm is able to reach mean-squared error as low as the Bayes-optimal error obtained when the model and its parameters are known and exactly matched in the inference procedure. Contents

https://hal-cea.archives-ouvertes.fr/cea-01933008/document

Approximate Survey Propagation for Statistical Inference

We study the random K-satisfiability problem using a partition functionwhere each solution is reweighted according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighted partition function. This allows us to obtain several new results on the properties of random K-satisfiability problem. In particular the reweighting allows to introduce a planted ensemble that generates instances that are, in some region of parameters, equivalent to random instances. We are hence able to generate at the same time a typical random SAT instance and one of its solutions. We study the relation between clustering and belief propagation fixed points and we give a direct evidence for the existence of purely entropic (rather than energetic) barriers between clusters in some region of parameters in the random K-satisfiability problem. We exhibit, in some large planted instances, solutions with a non-trivial whitening core; such solutions were known to exist but were so far never found on very large instances. Finally, we discuss algorithmic hardness of such planted instances and we determine a region of parameters in which planting leads to satisfiable benchmarks that, up to our knowledge, are the hardest known.

/pdf/reweighted-belief-propagation-and-quiet-planting-for-random-48xkcq1pnm.pdf

Reweighted Belief Propagation and Quiet Planting for Random K-SAT

An analysis of the SAT 2005 sub-competition on random instances is given This year this (sub-)competition set-up was geared to establish a basic setting, focusing on the instances near the (infamous) “50% densities”, so first we have to establish a more quantitative understanding of phase transitions here We present extended empirical results, clearly showing that the models used before, which are motivated by large-scale considerations, are inadequate at the relatively small scale considered here Then the series’ and all individual instances used in the competition are described We give a formal definition of the competition evaluation, and analyse the results of the competition, looking into the details of the scoring mechanism as well as into alternative evaluation methods

The SAT 2005 Solver Competition on Random Instances

Using the cavity equations of Mzard, Parisi, and Zecchina [Science 297 (2002), 812]; Mzard and Zecchina, [Phys Rev E 66 (2002), 056126] we derive the various threshold values for the number of clau...

Threshold values of random K-SAT from the cavity method

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Threshold values of random K-SAT from the cavity method