Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Glassy Behavior and Jamming of a Random Walk Process for Sequentially Satisfying a Constraint Satisfaction Formula

Random $K$-satisfiability ($K$-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random $K$-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process {\tt SEQSAT}, which satisfies a $K$-SAT formula in a sequential manner. Before satisfying each newly added clause, {\tt SEQSAT} walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density $\alpha$ of the satisfied subformula is less than certain value $\alpha_{cm}$; however it slows down considerably as $\alpha > \alpha_{cm}$ and finally reaches a jammed state at $\alpha \approx \alpha_{j}$. The glassy dynamical behavior of {\tt SEQSAT} for $\alpha \geq \alpha_{cm}$ probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point $\alpha_j$ is larger than the solution space clustering transition point $\alpha_d$, and its value can be predicted by a long-range frustration mean-field theory. For random $K$-SAT with $K\geq 4$, however, our simulation results indicate that $\alpha_j = \alpha_d$. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.Comment: 10 pages, 6 figures, 1 table, a mistake of numerical simulation corrected, and new results adde