Bicolouring random hypergraphs

Castellani, Tommaso; Napolano, Vincenzo; Ricci-Tersenghi, Federico; Zecchina, Riccardo

doi:10.1088/0305-4470/36/43/026

Cited by 18 publications

(27 citation statements)

References 18 publications

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“…The hypergraph bicoloring problem is NP-complete for K ≥ 3 [31]. The case K = 3 with a Poisson degree distribution for the variable nodes has already been studied in [32]. The authors found dynamical and SAT/UNSAT transitions within the single and multiple cluster approximations.…”

Section: Definition Of the Problem And Known Resultsmentioning

confidence: 99%

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

2008

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We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which types of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem--e.g., dynamical, rigidity, and satisfiability-unsatisfiability (SAT-UNSAT) transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance, we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so-called rigidity transition where the thermodynamically relevant clusters become frozen. These nonequilibrium solutions belong to the most probable unfrozen clusters.

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Section: Definition Of the Problem And Known Resultsmentioning

confidence: 99%

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

2008

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“…The satisfiability threshold is then given by the point l s where this entropy is zero. Second moment method in the balanced LOPs: In all the balanced LOPs that we have considered we found numerically that the second moment entropy, (20), is exactly twice the annealed entropy (16), 2s ann = s 2nd . A hint that this may happen comes from the following observations:…”

Section: The Balanced Lopsmentioning

confidence: 85%

Constraint satisfaction problems with isolated solutions are hard

Zdeborová¹,

Mézard²

2008

J. Stat. Mech.

110

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We study the phase diagram and the algorithmic hardness of the random 'locked' constraint satisfaction problems, and compare them to the commonly studied 'non-locked' problems like satisfiability of boolean formulas or graph coloring. The special property of the locked problems is that clusters of solutions are isolated points. This simplifies significantly the determination of the phase diagram, which makes the locked problems particularly appealing from the mathematical point of view. On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. Our results suggest that the easy/hard transition (for currently known algorithms) in the locked problems coincides with the clustering transition. These should thus be regarded as new benchmarks of really hard constraint satisfaction problems.

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“…A r = 0). For example A = (0, 1, 0, 0) corresponds to the positive 1-in-3 SAT [ZDEB-3], A = (0, 1, 1, 0) is bicoloring [CNRTZ03], A = (0, 1, 0, 1, 0) is 4-odd parity check (4-XOR-SAT without negations) [MRTZ03].…”

Section: Definitionmentioning

confidence: 99%

Statistical physics of hard optimization problems

Zdeborová¹

2009

Acta Physica Slovaca. Reviews and Tutorials

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Statistical physics of hard optimization problemsOptimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.

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Bicolouring random hypergraphs

Cited by 18 publications

References 18 publications

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

Constraint satisfaction problems with isolated solutions are hard

Statistical physics of hard optimization problems

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