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

Symmetry-breaking Answer Set Solving

In the context of Answer Set Programming, this paper investigates symmetry-breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We propose a reduction of disjunctive logic programs to a coloured digraph such that permutational symmetries can be constructed from graph automorphisms. Symmetries are then broken by introducing symmetry-breaking constraints. For this purpose, we formulate a preprocessor that integrates a graph automorphism system. Experiments demonstrate its computational impact.Comment: Proceedings of ICLP'10 Workshop on Answer Set Programming and Other Computing Paradig

Generalizing Boolean Satisfiability III: Implementation

This is the third of three papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high-performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal has been to define a representation in which this structure is apparent and can be exploited to improve computational performance. The first paper surveyed existing work that (knowingly or not) exploited problem structure to improve the performance of satisfiability engines, and the second paper showed that this structure could be understood in terms of groups of permutations acting on individual clauses in any particular Boolean theory. We conclude the series by discussing the techniques needed to implement our ideas, and by reporting on their performance on a variety of problem instances

Generalizing Boolean Satisfiability II: Theory

This is the second of three planned papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying ZAP, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the Davis-Putnam-Logemann-Loveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method. The third paper in this series discusses ZAPs implementation and presents experimental performance results

Symmetry Breaking for Answer Set Programming

In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry detection to a graph automorphism problem which allows to extract symmetries of a logic program from the symmetries of the constructed coloured graph. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process which implicitly represent symmetries and always with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. Furthermore, we explore symmetry breaking for answer set programming in two domains: first, constraint answer set programming as a novel approach to represent and solve constraint satisfaction problems, and second, distributed nonmonotonic multi-context systems. In particular, we formulate a translation-based approach to constraint answer set solving which allows for the application of our symmetry detection and symmetry breaking methods. To compare their performance with a-priori symmetry breaking techniques, we also contribute a decomposition of the global value precedence constraint that enforces domain consistency on the original constraint via the unit-propagation of an answer set solver. We evaluate both options in an empirical analysis. In the context of distributed nonmonotonic multi-context system, we develop an algorithm for distributed symmetry detection and also carry over symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201

The random K-satisfiability problem: from an analytic solution to an efficient algorithm

We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. 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. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the 3-sat problem, with very good performances.Comment: 38 pages, 13 figures; corrected typo

An adaptive prefix-assignment technique for symmetry reduction

This paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system. The technique is based on McKay's canonical extension framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the technique are (i) adaptability---the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries; (ii) parallelizability---prefix-assignments can be processed in parallel independently of each other; (iii) versatility---the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability---the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the technique and carry out a set of experiments that demonstrate ability to reduce symmetry on hard instances. Furthermore, we demonstrate that the implementation effectively parallelizes to compute clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie