Solution Techniques for Constraint Satisfaction Problems: Advanced Approaches

Handbook of Constraint Programming

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

We study two natural extensions of Constraint Satisfaction Problems (CSPs). Balance-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of Hard-Max-CSP consists of soft constraints and hard constraints, and the goal is to maximize the weight of satisfied soft constraints while satisfying all the hard constraints. These two extensions contain many fundamental problems not captured by CSPs, and challenge traditional theories about CSPs in a more general framework. Max-2-SAT and Max-Horn-SAT are the only two nontrivial classes of Boolean CSPs that admit a robust satisfibiality algorithm, i.e., an algorithm that finds an assignment satisfying at least (1 ? g(?)) fraction of constraints given a (1 ? ?)-satisfiable instance, where g(?) ? 0 as ? ? 0, and g(0) = 0. We prove the inapproximability of these problems with balance or hard constraints, showing that each variant changes the nature of the problems significantly (in different ways). For instance, deciding whether an instance of 2-SAT admits a balanced assignment is NP-hard, and for Max-2-SAT with hard constraints, it is hard to find a constant-factor approximation even on (1 ? ?)-satisfiable instances (in particular, the version with hard constraints does not admit a robust satisfiability algorithm). We also study hardness results for a certain CSP over a larger domain capturing ordering constraints: we show that hard constraints rule out constant-factor approximation algorithms. All our hardness results are almost optimal -- they completely rule out algorithms with certain properties, or can be matched by simple extensions to existing algorithms.

In this paper, Coevolutionary Genetic Algorithm for solving Constraint Satisfaction Problems (CSPs) is proposed. It consists of two Genetic Algorithms (GAs): a traditional GA and another GA to search for good schemata in the former GA. These GAs evolve in two levels, i.e., phenotype-level and schema-level, and affect with each other through genetic operations. To search for solutions effectively, we devise new genetic operator by utilizing search mechanism of solution synthesis approach used in CSP community. Computational results on general CSPs confirm the effectiveness of our approach.

An Iterative local-search framework for solving constraint satisfaction problem

Generalized arc-consistency propagation is predominantly used in constraint solvers to efficiently prune the search space when solving constraint satisfaction problems. Although many practical applications can be modelled as distributed constraint satisfaction problems, no distributed arc-consistency algorithms so far have considered the privacy of individual agents. In this paper, we propose a new distributed arc-consistency algorithm, called $$\mathsf {DisAC3.1}$$ , which leaks less private information of agents than existing distributed arc-consistency algorithms. In particular, $$\mathsf {DisAC3.1}$$ uses a novel termination determination mechanism, which allows the agents to share domains, constraints and communication addresses only with relevant agents. We further extend $$\mathsf {DisAC3.1}$$ to $$\mathsf {DisGAC3.1}$$ , which is the first distributed algorithm that enforces generalized arc-consistency on k-ary ( $$k\ge 2$$ ) constraint satisfaction problems. Theoretical analyses show that our algorithms are efficient in both time and space. Experiments also demonstrate that $$\mathsf {DisAC3.1}$$ outperforms the state-of-the-art distributed arc-consistency algorithm and that $$\mathsf {DisGAC3.1}$$ ’s performance scales linearly in the number of agents.

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called “islands of tractability.” A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language , whereas a constraint language is conservative if it contains all unary relations. Schaefer’s famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable constraint languages over a Boolean domain of values. Since then, many extensions and generalizations of this result have been obtained. Recently, Bulatov (TOCL 2011, JACM 2013) gave a full characterization of all islands of tractability for CSP and the counting version #CSP that are defined in terms of conservative constraint languages. This article addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages is not tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose, we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated, moves the instance to an island of tractability, that is, to a tractable class of instances. We consider strong backdoors into scattered classes , consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipelago of tractability . The main difficulty lies in finding a strong backdoor of given size k ; once it is found, we can try all possible instantiations of the backdoor variables and apply the polynomial time algorithms associated with the islands of tractability on the list component-wise. Our main result is an algorithm that, given a CSP instance with n variables, finds in time f ( k ) n O (1) a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size k or correctly decides that there is not such a backdoor. This also gives the running time for solving (#)CSP, provided that (#)CSP is polynomial-time tractable for the considered constraint languages. Our result makes significant progress towards the main goal of the backdoor-based approach to CSPs—the identification of maximal base classes for which small backdoors can be detected efficiently.

The University Course Timetabling Problem (UCTP) is a search problem that allocates a given number of rooms with given courses based on their scheduled time slots. UCTP belongs to the NP-complete class and can be defined as a constraint satisfaction problem (CSP). To solve the performance issue of CSP solvers, there are various local search methods. CSP solvers often use variable and value ordering heuristics to improve their search performance. Specific variable and value ordering heuristics can be even calculated by the help of a learning algorithm. Cluster-Specific Heuristics (CSH) are variable ordering heuristics which are learned based on clusters of CSPs. In this paper, to solve UCTP with user requirements, we propose a better performing CSH which is called Optimized Cluster-Specific Heuristics (OCSH). We have tested OCSH on generated time tabling problems with various user requirements and compared the runtime performances of variations of cluster-specific heuristics with OCSH. Finally, we show that OCSH is the best performing version of cluster specific heuristics to solve UCTP with user requirements.

We discuss a coevolutionary genetic algorithm for constraint satisfaction. Our basic idea is to explore effective genetic information in the population, i.e., schemata, and to exploit the genetic information in order to guide the population to better solutions. Our coevolutionary genetic algorithm (CGA) consists of two GA populations; the first GA, called searches for the solutions in a given environment (problem), and the second GA, called P-GA, searches for effective genetic information involved in the H-GA, namely, good schemata. Thus, each individual in P-GA consists of alleles in H-GA or don't care symbol representing a schema in the H-GA. These GA populations separately evolve in each genetic space at different abstraction levels and affect with each other by two genetic operators: superposition and transcription. We then applied our CGA to constraint satisfaction problems (CSPs) incorporating a new stochastic repair operator for P-GA to raise the consistency of schemata with the (local) constraint conditions in CSPs. We carried out two experiments: First, we examined the performance of CGA on various general CSPs that are generated randomly for a wide variety of density and tightness of constraint conditions in the CSPs that are the basic measures of characterizing CSPs. Next, we examined structured CSPs involving latent cluster structures among the variables in the CSPs. For these experiments, computer simulations confirmed us the effectiveness of our CGA.

The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure {\em adaptive width} and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case.

Overconstrained systems refer to sets of soft constraints which do not permit a solution satisfying all the constraints. The overconstrainedness in such soft-constraint systems can be manifested in a wide variety of structures, including weighted constraints, partially ordered constraints, constraint hierarchies and references. The simplest among these formalisms is the maximal constraint satisfaction problem (CSP), where a solution is sought which satisfies the maximum number of constraints. In this paper, backtracking algorithms and their intelligent versions used in the ordinary CSP context, are studied in context of the maximal CSP. The algorithms of E.C. Freuder and R.J. Wallace (1995) for depth-first branch-and-bound and backjumping are extended to conflict-directed backjumping. A theoretical analysis of the problem of application of intelligent backtracking algorithms for maximal CSP is provided.

Over-constrained problems are ubiquitous in real-world applications. In constraint programming, over-constrained problems can be modeled and solved using soft constraints. Soft constraints, as opposed to hard constraints, are allowed to be violated, and the goal is to find a solution that minimizes the total amount of violation. In this chapter, an overview of recent developments in solution methods for over-constrained problems using constraint programming is presented, with an emphasis on soft global constraints.

Constraint satisfaction arises in many domains in different forms. Search and inference compete for solving constraint satisfaction problems (CSPs) but the most successful approaches are those which benefit from both techniques. Based on this idea, this article introduces a new scheme for solving the general Max-CSP problem. The new approach exploits the simplicity and efficiency of a modified Particle Swarm Optimization and the advantage of adaptable inference levels offered by the Mini-Bucket Elimination algorithm. Experiments conducted on binary CSPs using different levels of inference are illustrative for the inference/search trade-off. Comparative studies highlight the differences between our stochastic population-based method and the systematic search performed by a Branch and Bound algorithm.

Brokerage between buyer and seller agents using Constraint Satisfaction Problem models

We perform a comprehensive study of mappings between constraint satisfaction problems (CSPs) and propositional satisfiability (SAT). We analyse four different mappings of SAT problems into CSPs, and two of CSPs into SAT problems. For each mapping, we compare the impact of achieving arc-consistency on the CSP with unit propagation on the SAT problem. We then extend these results to CSP algorithms that maintain (some level of) arc-consistency during search like FC and MAC, and to the Davis-Putnam procedure (which performs unit propagation at each search node). Because of differences in the branching structure of their search, a result showing the dominance of achieving arc-consistency on the CSP over unit propagation on the SAT problem does not necessarily translate to the dominance of MAC over the Davis-Putnam procedure. These results provide insight into the relationship between propositional satisfiability and constraint satisfaction.