Numerical Solution-Space Analysis of Satisfiability Problems

Abstract

The solution-space structure of the 3-Satisfiability Problem (3-SAT) is studied as a function of the control parameter alpha (ratio of number of clauses to the number of variables) using numerical simulations. For this purpose, one has to sample the solution space with uniform weight. It is shown here that standard stochastic local-search (SLS) algorithms like "ASAT" and "MCMCMC" (also known as "parallel tempering") exhibit a sampling bias. Nevertheless, unbiased samples of solutions can be obtained using the "ballistic-networking approach", which is introduced here. It is a generalization of "ballistic search" methods and yields also a cluster structure of the solution space. As application, solutions of 3-SAT instances are generated using ASAT plus ballistic networking. The numerical results are compatible with a previous analytic prediction of a simple solution-space structure for small values of alpha and a transition to a clustered phase at alpha_c ~ 3.86, where the solution space breaks up into several non-negligible clusters. Furthermore, in the thermodynamic limit there are, for values of alpha close to the SATUNSAT transition alpha_s ~ 4.267, always clusters without any frozen variables. This may explain why some SLS algorithms are able to solve very large 3-SAT instances close to the SAT-UNSAT transition.

12 pages, 14 figures

References in corpus (28)

• Exchange Monte Carlo Method and Application to Spin Glass Simulations
• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• The random K-satisfiability problem: from an analytic solution to an efficient algorithm
• Phase Transitions in the Coloring of Random Graphs
• Clustering of solutions in the random satisfiability problem
• The number of guards needed by a museum: A phase transition in vertex covering of random graphs
• The Entropy of the K-Satisfiability Problem
• A Landscape Analysis of Constraint Satisfaction Problems
• A variational description of the ground state structure in random satisfiability problems
• Clusters of solutions and replica symmetry breaking in random k-satisfiability
• Instability of one-step replica-symmetry-broken phase in satisfiability problems
• Core percolation in random graphs: a critical phenomena analysis
• Trajectories in phase diagrams, growth processes and computational complexity: how search algorithms solve the 3-Satisfiability problem
• Focused Local Search for Random 3-Satisfiability
• Generalization of the cavity method for adiabatic evolution of Gibbs states
• Circumspect descent prevails in solving random constraint satisfaction problems
• A practical guide to computer simulations
• Finding Low-Temperature States with Parallel Tempering, Simulated Annealing and Simple Monte Carlo
• Behavior of heuristics and state space structure near SAT/UNSAT transition
• Locked constraint satisfaction problems
• Exhaustive enumeration unveils clustering and freezing in random 3-SAT
• Spin Domains Generate Hierarchical Ground State Structure in J=+/-1 Spin Glasses
• Clustering analysis of the ground-state structure of the vertex-cover problem
• Communities of solutions in single solution clusters of a random K-Satisfiability formula
• A new method for analyzing ground-state landscapes: ballistic search
• Solution-space structure of (some) optimization problems
• Direct sampling of complex landscapes at low temperatures: the three-dimensional +/-J Ising spin glass
• Clustering of solutions in hard satisfiability problems

Cited by in corpus (11)

• Energy landscapes of combinatorial optimization in Ising machines
• Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
• Replica symmetry breaking for Ulam's problem
• Counting solutions from finite samplings
• Adding color: Visualization of energy landscapes in spin glasses
• Replica-symmetry breaking for directed polymers
• Optimal Vertex Cover for the Small-World Hanoi Networks
• Phase transition for parameter learning of Hidden Markov Models
• Computing a Knot Invariant as a Constraint Satisfaction Problem
• Phase transition in the bipartite z-matching
• Finite-size scaling in random $K$-satisfiability problems