381 research outputs found
Strong Refutation Heuristics for Random k-SAT
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.A simple first moment argument shows that in a randomly chosen -SAT formula with clauses over boolean variables, the fraction of satisfiable clauses is as almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random -SAT formula, can we find a certificate that the fraction of satisfiable clauses is in polynomial time? We present heuristics based on spectral techniques that in the case and , and in the case and , find such certificates almost surely. In addition, we present heuristics for bounding the independence number (resp. the chromatic number) of random -uniform hypergraphs from above (resp. from below) for .Peer Reviewe
Going after the k-SAT Threshold
Random -SAT is the single most intensely studied example of a random
constraint satisfaction problem. But despite substantial progress over the past
decade, the threshold for the existence of satisfying assignments is not known
precisely for any . The best current results, based on the second
moment method, yield upper and lower bounds that differ by an additive , a term that is unbounded in (Achlioptas, Peres: STOC 2003).
The basic reason for this gap is the inherent asymmetry of the Boolean value
`true' and `false' in contrast to the perfect symmetry, e.g., among the various
colors in a graph coloring problem. Here we develop a new asymmetric second
moment method that allows us to tackle this issue head on for the first time in
the theory of random CSPs. This technique enables us to compute the -SAT
threshold up to an additive . Independently of
the rigorous work, physicists have developed a sophisticated but non-rigorous
technique called the "cavity method" for the study of random CSPs (M\'ezard,
Parisi, Zecchina: Science 2002). Our result matches the best bound that can be
obtained from the so-called "replica symmetric" version of the cavity method,
and indeed our proof directly harnesses parts of the physics calculations
The asymptotics of the clustering transition for random constraint satisfaction problems
Random Constraint Satisfaction Problems exhibit several phase transitions
when their density of constraints is varied. One of these threshold phenomena,
known as the clustering or dynamic transition, corresponds to a transition for
an information theoretic problem called tree reconstruction. In this article we
study this threshold for two CSPs, namely the bicoloring of -uniform
hypergraphs with a density of constraints, and the -coloring of
random graphs with average degree . We show that in the large limit
the clustering transition occurs for , , where is the same constant for both models. We
characterize via a functional equation, solve the latter
numerically to estimate , and obtain an analytic
lowerbound . Our
analysis unveils a subtle interplay of the clustering transition with the
rigidity (naive reconstruction) threshold that occurs on the same asymptotic
scale at .Comment: 35 pages, 8 figure
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
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