Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems

  title={Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems},
  author={Florent Krzakala and Lenka Zdeborov{\'a}},
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. 
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    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
An exact analytical expression for the two-coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges are presented.
A Landscape Analysis of Constraint Satisfaction Problems
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  • Mathematics, Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2007
An analysis of constraint satisfaction problems, such as sphere packing, K-SAT, and graph coloring, in terms of an effective energy landscape is discussed, to better characterize the J-point, proposed as a systematic definition of random close packing, and to place it in the context of other theories of glasses.
Rigorous location of phase transitions in hard optimization problems
The results prove that the heuristic predictions of statistical physics in this context are essentially correct and establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.
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A general criterion for the validity of this ansatz is derived and evidence that the 1RSB solution gives exact threshold values c(q) for the transition from the colorable to the uncolorable phase with q colors is provided.
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The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample.