A Partial Taxonomy of Substitutability and Interchangeability

Substitutability, interchangeability and related concepts in Constraint Programming were introduced approximately twenty years ago and have given rise to considerable subsequent research. We survey this work, classify, and relate the different concepts, and indicate directions for future work, in particular with respect to making connections with research into symmetry breaking. This paper is a condensed version of a larger work in progress.Comment: 18 pages, The 10th International Workshop on Symmetry in Constraint Satisfaction Problems (SymCon'10

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

Improving the Computational Efficiency in Symmetrical Numeric Constraint Satisfaction Problems

Models are used in science and engineering for experimentation, analysis, diagnosis or design. In some cases, they can be considered as numeric constraint satisfaction problems (NCSP). Many models are symmetrical NCSP. The consideration of symmetries ensures that NCSP-solver will find solutions if they exist on a smaller search space. Our work proposes a strategy to perform it. We transform the symmetrical NCSP into a newNCSP by means of addition of symmetry-breaking constraints before the search begins. The specification of a library of possible symmetries for numeric constraints allows an easy choice of these new constraints. The summarized results of the studied cases show the suitability of the symmetry-breaking constraints to improve the solving process of certain types of symmetrical NCSP. Their possible speedup facilitates the application of modelling and solving larger and more realistic problems.Ministerio de Ciencia y Tecnología DIP2003-0666-02-

Elimination des symétries locales durant la résolution dans les CSPs

Plusieurs approches exploitant l'élimination des symétries dans la résolution des CSPs sont apparues récemment. La grande majorité de ces méthodes exploitent les symétries globales du problème étudié et ne tente pas d'exploiter les symétries locales. Il a été montré que l'élimination des symétries globales peut être utile dans la résolution des CSPs. Mais exploiter uniquement ces symétries peut ne pas suffire pour résoudre des problèmes difficiles contenant de nombreuses symétries locales. En effet, un problème peut avoir peu ou pas du tout de symétries initiales (globales) et devenir très symétrique à certains noeuds durant la recherche. Dans ce papier, nous étudions le principe général de la symétrie sémantique et on définit la symétrie syntaxique qui est une condition suffisante de la symétrie sémantique. Nous montrons comment la symétrie syntaxique est détectée et éliminée localement pour améliorer l'efficacité des méthodes de résolution de CSPs. Les expérimentations confirment que l'exploitation des symétries locales est profitable dans la résolution des CSPs

Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure

Decentralized Constraint Satisfaction

We show that several important resource allocation problems in wireless networks fit within the common framework of Constraint Satisfaction Problems (CSPs). Inspired by the requirements of these applications, where variables are located at distinct network devices that may not be able to communicate but may interfere, we define natural criteria that a CSP solver must possess in order to be practical. We term these algorithms decentralized CSP solvers. The best known CSP solvers were designed for centralized problems and do not meet these criteria. We introduce a stochastic decentralized CSP solver and prove that it will find a solution in almost surely finite time, should one exist, also showing it has many practically desirable properties. We benchmark the algorithm's performance on a well-studied class of CSPs, random k-SAT, illustrating that the time the algorithm takes to find a satisfying assignment is competitive with stochastic centralized solvers on problems with order a thousand variables despite its decentralized nature. We demonstrate the solver's practical utility for the problems that motivated its introduction by using it to find a non-interfering channel allocation for a network formed from data from downtown Manhattan