Message passing for vertex covers

Weigt, Martin; Zhou, Haijun

doi:10.1103/physreve.74.046110

Cited by 52 publications

(119 citation statements)

References 32 publications

(63 reference statements)

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“…and 2 ( ) has to be determined through (33). Equation (53) which is stable up to , as already noticed.…”

Section: G ( )mentioning

confidence: 85%

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The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

Lucibello

Ricci-Tersenghi

2014

International Journal of Statistical Mechanics

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We analyse the asymptotic behaviour of random instances of the maximum set packing (MSP) optimization problem, also known as maximum matching or maximum strong independent set on hypergraphs. We give an analytic prediction of the MSPs size using the 1RSB cavity method from statistical mechanics of disordered systems. We also propose a heuristic algorithm, a generalization of the celebrated Karp-Sipser one, which allows us to rigorously prove that the replica symmetric cavity method prediction is exact for certain problem ensembles and breaks down when a core survives the leaf removal process. The -phenomena threshold discovered by Karp and Sipser, marking the onset of core emergence and of replica symmetry breaking, is elegantly generalized to = /( −1) for one of the ensembles considered, where is the size of the sets.

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“…and 2 ( ) has to be determined through (33). Equation (53) which is stable up to , as already noticed.…”

Section: G ( )mentioning

confidence: 85%

“…Regarding the problem of looking for optimal solutions in single samples, we believe that an efficient heuristic algorithm, able to obtain near-to-optimal configurations also in the RSB phase, could be constructed following the ideas of [33].…”

Section: Discussionmentioning

confidence: 99%

The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

Lucibello

Ricci-Tersenghi

2014

International Journal of Statistical Mechanics

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“…They are "Vercov" [3], "warning" [42], and genetic algorithm (GA) [43]- [45]. The Vercov method is a relatively simple yet well-known benchmark algorithm requiring no global information.…”

Section: Comparison With Other Algorithmsmentioning

confidence: 99%

Towards a Snowdrift Game Optimization to Vertex Cover of Networks

Yang

2013

IEEE Trans. Cybern.

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To solve the vertex cover problem in an agent-based and distributed networking systems utilizing local information, we treat each vertex as an intelligent rational agent rather than an inanimate one and provide a spatial-snowdrift-game-based optimization framework to vertex cover of networks. We analyze the inherent relation between the snowdrift game and the vertex cover: Strict Nash equilibriums of the spatial snowdrift game are the intermediate states between vertex-covered and minimal-vertex-covered states. Such equilibriums are obtained by employing the memory-based best response update rule. We also find that a better approximate solution in terms of the minimal vertex cover will be achieved by increasing the individuals' memory length, because such a process optimizes the individuals' strategies and helps them convert from bad equilibriums into better ones. Our findings pave a new way to solve the vertex cover problem from the perspective of agent-based self-organized optimization.

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“…One has to extend the theory to infinite steps of replica symmetry breaking to get a marginally stable mean-field solution. However, experiences obtained on the minimal vertex cover problem [23] and other combinatorial optimization problems [21] have indicated that even though the 1RSB mean-field solution is not stable, the ground-state energy density predicted by this solution is actually very close to the true value. We therefore believe that the crossing point of the two curves of Fig.…”

Section: The Reduction Of Sparse Random 3-sat To 3-xorsatmentioning

confidence: 99%

Witness of unsatisfiability for a random 3-satisfiability formula

Zhou

Alava

et al. 2013

Phys. Rev. E

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The random 3-satisfiability (3-SAT) problem is in the unsatisfiable (UNSAT) phase when the clause density α exceeds a critical value α s ≈ 4.267. Rigorously proving the unsatisfiability of a given large 3-SAT instance is, however, extremely difficult. In this paper we apply the mean-field theory of statistical physics to the unsatisfiability problem, and show that a reduction to 3-XORSAT, which permits the construction of a specific type of UNSAT witnesses (Feige-Kim-Ofek witnesses), is possible when the clause density α > 19. We then construct Feige-Kim-Ofek witnesses for single 3-SAT instances through a simple random sampling algorithm and a focused local search algorithm. The random sampling algorithm works only when α scales at least linearly with the variable number N , but the focused local search algorithm works for clause density α > cN b with b ≈ 0.59 and prefactor c ≈ 8. The exponent b can be further decreased by enlarging the single parameter S of the focused local search algorithm.

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Message passing for vertex covers

Cited by 52 publications

References 32 publications

The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

Towards a Snowdrift Game Optimization to Vertex Cover of Networks

Witness of unsatisfiability for a random 3-satisfiability formula

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