35 research outputs found
Catching the k-NAESAT Threshold
The best current estimates of the thresholds for the existence of solutions
in random constraint satisfaction problems ('CSPs') mostly derive from the
first and the second moment method. Yet apart from a very few exceptional cases
these methods do not quite yield matching upper and lower bounds. According to
deep but non-rigorous arguments from statistical mechanics, this discrepancy is
due to a change in the geometry of the set of solutions called condensation
that occurs shortly before the actual threshold for the existence of solutions
(Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). To
cope with condensation, physicists have developed a sophisticated but
non-rigorous formalism called Survey Propagation (Mezard, Parisi, Zecchina:
Science 2002). This formalism yields precise conjectures on the threshold
values of many random CSPs. Here we develop a new Survey Propagation inspired
second moment method for the random k-NAESAT problem, which is one of the
standard benchmark problems in the theory of random CSPs. This new technique
allows us to overcome the barrier posed by condensation rigorously. We prove
that the threshold for the existence of solutions in random -NAESAT is
2^{k-1}\ln2-(\frac{\ln2}2+\frac14)+\eps_k, where |\eps_k| \le
2^{-(1-o_k(1))k}, thereby verifying the statistical mechanics conjecture for
this problem
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
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure