Instances Of Constraint Satisfaction Problems Research Articles

Symmetry Definitions for Constraint Satisfaction Problems

We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.

Rigorous location of phase transitions in hard optimization problems

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme 'hardness', computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

Tractable Decision for a Constraint Language Implies Tractable Search

A constraint satisfaction problem (CSP) instance has a set of variables, each of which can take values in some domain. It also has a set of constraints, each of which restricts the variables in its scope to take values limited by its constraint relation. The language of a constraint satisfaction problem instance is the set of different constraint relations used in its specification. For a given set of relations Γ over some domain we define the problem CSP (Γ) to the set of CSP instances whose language is contained in Γ. The decision problem for a set of CSP instances is, given an instance in the class, to decide whether a solution exists. The search problem is to find such a solution. Here we address the connection between the tractability of the decision and search problems. We prove that given a constraint language Γ over a finite domain for which the decision problem for CSP (Γ) is tractable, the search problem is always tractable. We define a surjective language over a finite domain in a standard way. We also show that we can determine in polynomial time whether an instance over a surjective language with a tractable decision problem has fewer than k solutions, and that we can generate all of its solutions with polynomial delay.

Property testers for dense constraint satisfaction programs on finite domains

AbstractMany NP‐hard languages can be “decided” in subexponential time if the definition of “decide” is relaxed only slightly. Rubinfeld and Sudan introduced the notion of property testers, probabilistic algorithms that can decide, with high probability, if a function has a certain property or if it is far from any function having this property. Goldreich, Goldwasser, and Ron constructed property testers with constant query complexity for dense instances of a large class of graph problems. Since many graph problems can be viewed as special cases of the Constraint Satisfaction Problem on Boolean domains, it is natural to try to construct property testers for more general cases of the Constraint Satisfaction Problem. In this paper, we give explicit constructions of property testers using a constant number of queries for dense instances of Constraint Satisfaction Problems where the constraints have constant arity and the variables assume values in some domain of finite size. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 14–32, 2002

GA performance distributions and randomly generated binary constraint satisfaction problems

We investigate the variable performance of a genetic algorithm (GA) on randomly generated binary constraint satisfaction problem instances which occur near the phase transition from soluble to non-soluble. We first carry out a conventional landscape analysis and observe, next to a number of common features related to the interaction structure, important differences between the instances, such as the number of solutions, the quality of the paths to the solutions, and the lengths and extent of the neutral paths for mutation. We then split the dynamics of the GA into two phases: the ascent towards the high fitness region, and from this high fitness region to a solution. To gain further information about the GA's behavior in the first phase, we construct two models based on the much simpler fully separable functions, and try to match instances which show a similar performance distribution. Although far from exact, this technique of comparing with analog search problems gives a hint about the landscape elements that are responsible for the GA's slow ascent.

Constraint Propagation in Graph Coloring

In this paper we propose a method for integrating constraint propagation algorithms into an optimization procedure for vertex coloring with the goal of finding improved lower bounds. The key point we address is how to get instances of Constraint Satisfaction Problems (CSPs) from a graph coloring problem in order to give rise to new lower bounds outperforming the maximum clique bound. More precisely, the algorithms presented have the common goal of finding CSPs in the graph for which infeasibility can be proven. This is achieved by means of constraint propagation techniques which allow the algorithms to eliminate inconsistencies in the CSPs by updating domains dynamically and rendering such infeasibilities explicit. At the end of this process we use the largest CSP for which it has not been possible to prove infeasibility as an input for an algorithm which enlarges such CSP to get a feasible coloring. We experimented with a set of middle-high density graphs with quite a large difference between the maximum clique and the chromatic number.

A Lagrangian reconstruction of GENET

GENET is a heuristic repair algorithm which demonstrates impressive efficiency in solving some large-scale and hard instances of constraint satisfaction problems (CSPs). In this paper, we draw a surprising connection between GENET and discrete Lagrange multiplier methods. Based on the work of Wah and Shang, we propose a discrete Lagrangian-based search scheme LSDL , defining a class of search algorithms for solving CSPs. We show how GENET can be reconstructed from LSDL . The dual viewpoint of GENET as a heuristic repair method and a discrete Lagrange multiplier method allows us to investigate variants of GENET from both perspectives. Benchmarking results confirm that first, our reconstructed GENET has the same fast convergence behavior as the original GENET implementation, and has competitive performance with other local search solvers DLM, WalkSAT, and W sat( oip), on a set of difficult benchmark problems. Second, our improved variant, which combines techniques from heuristic repair and discrete Lagrangian methods, is always more efficient than the reconstructed GENET, and can better it by an order of magnitude.

IMPROVING EVOLUTIONARY ALGORITHMS FOR EFFICIENT CONSTRAINT SATISFACTION

Hard or large-scale constraint satisfaction and optimization problems, occur widely in artificial intelligence and operations research. These problems are often difficult to solve with global search methods, but many of them can be efficiently solved by local search methods. Evolutionary algorithms are local search methods which have considerable success in tackling difficult, or ill-defined optimization problems. In contrast they have not been so successful in tackling constraint satisfaction problems. Other local search methods, in particular GENET and EGENET are designed specifically for constraint satisfaction problems, and have demonstrated remarkable success in solving hard examples of these problems. In this paper we examine how we can transfer the mechanisms that were so successful in (E)GENET to evolutionary algorithms, in order to tackle constraint satisfaction algorithms efficiently. An empirical comparison of our evolutionary algorithm improved by mechanisms from EGENET and shows how it can markedly improve on the efficiency of EGENET in solving certain hard instances of constraint satisfaction problems.

Constraints and universal algebra

In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed by a constraint satisfaction problem with a given structure, or a given set of allowed constraint types.

An exact algorithm for the constraint satisfaction problem: Application to logical inference

The inference problem in propositional logic realizes a strong connection between Artificial Intelligence and Operational Research. It is now well-known that this problem can be formulated as a constraint satisfaction problem (CSP), whose system has a generalized covering type (we recall that a CSP consists in proving the emptiness of a domain defined by a set of diophantine constraints, or the existence of a solution). We propose a new method - denoted by FAST (Fast Algorithm for the constraint Satisfaction Test problems) - which allows an efficient solution of the CSP instances for logical inference. Computational results are reported for 3-SAT, 4-SAT and system expert type instances.