scholarly journals On the solution-space geometry of random constraint satisfaction problems

2011 ◽  
Vol 38 (3) ◽  
pp. 251-268 ◽  
Author(s):  
Dimitris Achlioptas ◽  
Amin Coja-Oghlan ◽  
Federico Ricci-Tersenghi
Keyword(s):  
Constraint Satisfaction ◽  
Solution Space ◽  
Constraint Satisfaction Problems ◽  
Space Geometry ◽  
Random Constraint Satisfaction Problems

scholarly journals CRITICALITY AND HETEROGENEITY IN THE SOLUTION SPACE OF RANDOM CONSTRAINT SATISFACTION PROBLEMS

2010 ◽  
Vol 24 (18) ◽  
pp. 3479-3487 ◽  
Author(s):  
HAIJUN ZHOU
Keyword(s):  
Constraint Satisfaction ◽  
Spin Glasses ◽  
Solution Space ◽  
Threshold Value ◽  
Critical Value ◽  
Constraint Satisfaction Problems ◽  
Model Systems ◽  
Dynamical Heterogeneity ◽  
Ergodicity Breaking ◽  
Random Constraint Satisfaction Problems

Random constraint satisfaction problems are interesting model systems for spin-glasses and glassy dynamics studies. As the constraint density of such a system reaches certain threshold value, its solution space may split into extremely many clusters. In this work we argue that this ergodicity-breaking transition is preceded by a homogeneity-breaking transition. For random K-SAT and K-XORSAT, we show that many solution communities start to form in the solution space as the constraint density reaches a critical value αcm, with each community containing a set of solutions that are more similar with each other than with the outsider solutions. At αcm the solution space is in a critical state. The connection of these results to the onset of dynamical heterogeneity in lattice glass models is discussed.

scholarly journals Constraint Satisfaction by Survey Propagation

2005 ◽  
Author(s):  
Alfredo Braunstein ◽  
Marc Mézard
Keyword(s):  
Constraint Satisfaction ◽  
Message Passing ◽  
Statistical Physics ◽  
Belief Propagation ◽  
Solution Space ◽  
Error Correcting Codes ◽  
Constraint Satisfaction Problems ◽  
Probabilistic Methods ◽  
Survey Propagation ◽  
Algorithmic Strategy

Methods and analyses from statistical physics are of use not only in studying the performance of algorithms, but also in developing efficient algorithms. Here, we consider survey propagation (SP), a new approach for solving typical instances of random constraint satisfaction problems. SP has proven successful in solving random k-satisfiability (k -SAT) and random graph q-coloring (q-COL) in the “hard SAT” region of parameter space [79, 395, 397, 412], relatively close to the SAT/UNSAT phase transition discussed in the previous chapter. In this chapter we discuss the SP equations, and suggest a theoretical framework for the method [429] that applies to a wide class of discrete constraint satisfaction problems. We propose a way of deriving the equations that sheds light on the capabilities of the algorithm, and illustrates the differences with other well-known iterative probabilistic methods. Our approach takes into account the clustered structure of the solution space described in chapter 3, and involves adding an additional “joker” value that variables can be assigned. Within clusters, a variable can be frozen to some value, meaning that the variable always takes the same value for all solutions (satisfying assignments) within the cluster. Alternatively, it can be unfrozen, meaning that it fluctuates from solution to solution within the cluster. As we will discuss, the SP equations manage to describe the fluctuations by assigning joker values to unfrozen variables. The overall algorithmic strategy is iterative and decomposable in two elementary steps. The first step is to evaluate the marginal probabilities of frozen variables using the SP message-passing procedure. The second step, or decimation step, is to use this information to fix the values of some variables and simplify the problem. The notion of message passing will be illustrated throughout the chapter by comparing it with a simpler procedure known as belief propagation (mentioned in ch. 3 in the context of error correcting codes) in which no assumptions are made about the structure of the solution space. The chapter is organized as follows. In section 2 we provide the general formalism, defining constraint satisfaction problems as well as the key concepts of factor graphs and cavities, using the concrete examples of satisfiability and graph coloring.

The replica symmetric phase of random constraint satisfaction problems

2019 ◽  
Vol 29 (3) ◽  
pp. 346-422
Author(s):  
Amin Coja-Oghlan ◽  
Tobias Kapetanopoulos ◽  
Noela Müller
Keyword(s):  
Phase Transition ◽  
Constraint Satisfaction ◽  
Broad Class ◽  
Constraint Satisfaction Problems ◽  
Precise Location ◽  
Combinatorial Structures ◽  
Substantial Impact ◽  
Long Range Correlations ◽  
Symmetric Phase ◽  
Random Constraint Satisfaction Problems

AbstarctRandom constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications for Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g. Banks et al. 2016, Proc. 29th COLT).

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