1,618 research outputs found
The condensation phase transition in random graph coloring
Based on a non-rigorous formalism called the "cavity method", physicists have
put forward intriguing predictions on phase transitions in discrete structures.
One of the most remarkable ones is that in problems such as random -SAT or
random graph -coloring, very shortly before the threshold for the existence
of solutions there occurs another phase transition called "condensation"
[Krzakala et al., PNAS 2007]. The existence of this phase transition appears to
be intimately related to the difficulty of proving precise results on, e.g.,
the -colorability threshold as well as to the performance of message passing
algorithms. In random graph -coloring, there is a precise conjecture as to
the location of the condensation phase transition in terms of a distributional
fixed point problem. In this paper we prove this conjecture for exceeding a
certain constant
The condensation transition in random hypergraph 2-coloring
For many random constraint satisfaction problems such as random
satisfiability or random graph or hypergraph coloring, the best current
estimates of the threshold for the existence of solutions are based on the
first and the second moment method. However, in most cases these techniques do
not yield matching upper and lower bounds. Sophisticated but non-rigorous
arguments from statistical mechanics have ascribed this discrepancy to the
existence of a phase transition called condensation that occurs shortly before
the actual threshold for the existence of solutions and that affects the
combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi,
Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time
that a condensation transition exists in a natural random CSP, namely in random
hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment
method breaks down strictly \emph{before} the condensation transition. Our
proof also yields slightly improved bounds on the threshold for random
hypergraph 2-colorability. We expect that our techniques can be extended to
other, related problems such as random k-SAT or random graph k-coloring
Solution space structure of random constraint satisfaction problems with growing domains
In this paper we study the solution space structure of model RB, a standard
prototype of Constraint Satisfaction Problem (CSPs) with growing domains. Using
rigorous the first and the second moment method, we show that in the solvable
phase close to the satisfiability transition, solutions are clustered into
exponential number of well-separated clusters, with each cluster contains
sub-exponential number of solutions. As a consequence, the system has a
clustering (dynamical) transition but no condensation transition. This picture
of phase diagram is different from other classic random CSPs with fixed domain
size, such as random K-Satisfiability (K-SAT) and graph coloring problems,
where condensation transition exists and is distinct from satisfiability
transition. Our result verifies the non-rigorous results obtained using cavity
method from spin glass theory, and sheds light on the structures of solution
spaces of problems with a large number of states.Comment: 8 pages, 1 figure
Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems
We review the understanding of the random constraint satisfaction problems,
focusing on the q-coloring of large random graphs, that has been achieved using
the cavity method of the physicists. We also discuss the properties of the
phase diagram in temperature, the connections with the glass transition
phenomenology in physics, and the related algorithmic issues.Comment: 10 pages, Proceedings of the International Workshop on
Statistical-Mechanical Informatics 2007, Kyoto (Japan) September 16-19, 200
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are
spatially coupled across a finite window along the chain direction. We
investigate their phase diagram at zero temperature using the survey
propagation formalism and the interpolation method. We prove that the SAT-UNSAT
phase transition threshold of an infinite chain is identical to the one of the
individual standard model, and is therefore not affected by spatial coupling.
We compute the survey propagation complexity using population dynamics as well
as large degree approximations, and determine the survey propagation threshold.
We find that a clustering phase survives coupling. However, as one increases
the range of the coupling window, the survey propagation threshold increases
and saturates towards the phase transition threshold. We also briefly discuss
other aspects of the problem. Namely, the condensation threshold is not
affected by coupling, but the dynamic threshold displays saturation towards the
condensation one. All these features may provide a new avenue for obtaining
better provable algorithmic lower bounds on phase transition thresholds of the
individual standard model
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