On the Analysis of Backtrack Proceduresfor the Colouring of Random Graphs

Monasson, Rémi

doi:10.1007/978-3-540-44485-5_11

Cited by 16 publications

(13 citation statements)

References 15 publications

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“…There are many other results, known for dll, that can be easily shown to apply to cmodels, and thanks to our results, could be applicable to smodels and dlv algorithms, e.g., [44] about the average complexity of coloring randomly generated graphs, and [1] on lower bounds on random 3-CNF formulas also for densities significantly below the satisfiability threshold d ≈ 4.23.…”

Section: Proposition 6 the Complexity Of Cmodels Smodels And DLV On mentioning

confidence: 57%

On the relation among answer set solvers

Giunchiglia

Leone

Maratea

2008

Ann Math Artif Intell

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In this paper, we study the relation among Answer Set Programming (ASP) systems from a computational point of view. We consider smodels, dlv, and cmodels ASP systems based on stable model semantics, the first two being native ASP systems and the last being a SAT-based system. We first show that smodels, dlv, and cmodels explore search trees with the same branching nodes (assuming, of course, a same branching heuristic) on the class of tight logic programs. Leveraging on the fact that SAT-based systems rely on the deeply studied Davis-LogemannLoveland (dll) algorithm, we derive new complexity results for the ASP procedures. We also show that on nontight programs the SAT-based systems are computationally different from native procedures, and the latter have computational advantages. Moreover, we show that native procedures can guarantee the "correctness" of a reported solution when reaching the leaves of the search trees (i.e., no stability check is needed), while this is not the case for SAT-based procedures on nontight programs. A similar advantage holds for dlv in comparison with smodels if the "well-founded" operator is disabled and only Fitting's operator is used for negative inferences. We finally study the "cost" of achieving such advantages and comment on to what extent the results presented extend to other systems.A preliminary version of a part of the results presented in this work has been published in [25].

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Section: Proposition 6 the Complexity Of Cmodels Smodels And DLV On mentioning

confidence: 57%

On the relation among answer set solvers

Giunchiglia

Leone

Maratea

2008

Ann Math Artif Intell

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“…After first results on the satisfiability problem [44], this machinery was also used to study the 3-coloring problem. In particular, Monasson and co-workers modeled the solution process of backtrack search with an out-of-equilibrium (multi-dimensional) surface growth problem [45,31]. By solving the resulting partial differential equation, an estimation of the backtrack algorithm's runtime can be obtained that is fairly close to the empirical results for relatively dense graphs.…”

Section: Previous Workmentioning

confidence: 86%

“…In the under-constrained case, coloring is easy: even the simplest heuristics usually find a proper coloring [30,24]. In the over-constrained case, it is easy for backtracking algorithms to prove uncolorability because they quickly reach contradiction [31].…”

Section: Previous Workmentioning

confidence: 99%

Average-case complexity of backtrack search for coloring sparse random graphs

Mann

Szajkó

2013

Journal of Computer and System Sciences

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We investigate asymptotically the expected number of steps taken by backtrack search for k-coloring random graphs G n,p(n) or proving non-k-colorability, where p(n) is an arbitrary sequence tending to 0, and k is constant. Contrary to the case of constant p, where the expected runtime is known to be O(1), we prove that here the expected runtime tends to infinity. We establish how the asymptotic behaviour of the expected number of steps depends on the sequence p(n). In particular, for p(n) = d/n, where d is a constant, the runtime is always exponential, but it can be also polynomial if p(n) decreases sufficiently slowly, e.g. for p(n) = 1/ ln n.

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“…Our approach has already been extended to other decision problems, e.g. the vertex covering of random graphs (Hartmann and Weigt, 2001) or the coloring of random graphs (Ein-Dor and Monasson, 2003) (see (Jia and Moore, 2003) for recent rigorous results on backtracking in this case). It is important to stress that it is not limited to the determination of the average solving time, but may also be used to capture its distribution (Gent and Walsh, 1994;Cocco and Monasson, 2002;Montanari and Zecchina, 2002) and to understand the efficiency of restarts techniques (Gomes et al, 2000).…”

Section: Discussionmentioning

confidence: 99%

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

Monasson¹

2004

Theoretical Computer Science

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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 α 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 α 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 ω with ω > 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 ω as a function of α. 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.

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On the Analysis of Backtrack Proceduresfor the Colouring of Random Graphs

Cited by 16 publications

References 15 publications

On the relation among answer set solvers

On the relation among answer set solvers

Average-case complexity of backtrack search for coloring sparse random graphs

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

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