Soft Directional Substitutable based Decompositions for MOVCSP

2008

Preferences in constraint problems are common but significant in many real world applications. In this paper, we extend our conditional and composite CSP (CCCSP) framework, managing CSPs in a dynamic environment, in order to handle preferences. Unlike the existing CSP models managing one form of preferences, ours supports four types, namely : variable value and constraint preferences, composite preferences and conditional preferences. This offers more expressive power in representing a wide variety of constraint problems. The preferences are considered here as a set of soft constraints using a c-semiring structure with combination and projection operators. Solving constraint problems with preferences consists of finding a solution satisfying all the constraints while optimizing the preference values. This is handled by a variant of the branch and bound algorithm, we propose in this paper, and where constraint propagation is used to improve the time efficiency. Experimental tests, we conducted on randomly generated CCCSPs with preferences, favor the MAC principle as the constraint propagation strategy to be used within the branch and bound procedure.

Lecture Notes in Computer Science, 2003

In this article, we propose a new soft constraint called preference constraint, squaring well with the decision theory concept of preference binary relation. We show how to use it for designing complex hierarchical preference information based on preference binary relations for combinatorial problems. Finally, preference-based constraint systems are defined and associated best quality choice problems are introduced. This new model offers greater flexibility to represent and make complex decisions with computers.

2001

Abstract. Many constraint satisfaction problems (csp's) are formulated with 0/1 variables. Sometimes this is a natural encoding, sometimes it is as a result of a reformulation of the problem, other times 0/1 variables make up only a part of the problem. Frequently we have constraints that restrict the sum of the values of variables. This can be encoded as a simple summation of the variables. However, since variables can only take 0/1 values we can also use an occurrence constraint, eg the number of occurrences of 1 must be k.

Information Sciences, 2013

ABSTRACT Signed clausal forms offer a suitable logical framework for automated reasoning in multiple-valued logics. It turns out that the satisfiability problem of any finitely-valued propositional logic, as well as of certain infinitely-valued logics, can be easily reduced, in polynomial time, to the satisfiability problem of signed clausal forms. On the other hand, signed clausal forms are a powerful knowledge representation language for constraint programming, and have shown to be a practical and competitive approach to solving combinatorial decision problems.Motivated by the existing theoretical and practical results for the satisfiability problem of signed clausal forms, as well as by the recent logical and algorithmic results on the Boolean maximum satisfiability problem, in this paper we investigate the maximum satisfiability problem of propositional signed clausal forms from the logical and practical points of view. From the logical perspective, our aim is to define complete inference systems taking as a starting point the resolution-style calculi defined for the Boolean CNF case. The result is the definition of two sound and complete resolution-style rules, called signed binary resolution and signed parallel resolution for maximum satisfiability. From the practical perspective, our main motivation is to use the language of signed clausal forms along with the newly defined inference systems as a generic approach to solve combinatorial optimization problems, and not just for solving decision problems as so far. The result is the establishment of a link between signed logic and constraint programming that provides a concise and elegant logical framework for weighted constraint programming.

2006

Abstract Set variables are ubiquitous in modeling (soft) constraint problems, but efforts on practical consistency algorithms for Weighted Constraint Satisfaction Problems (WCSPs) have only been on integer variables. We adapt the classical notion of set bounds consistency for WC-SPs, and propose efficient representation schemes for set variables and common unary, binary, and ternary set constraints, as well as cardinality constraints.

Successful mixed integer nonlinear programming algorithms rely on computing tight over-and underestimators for nonlinear functions. With the advances of mixed integer linear programming, polyhedral approximations over subregions of the original feasible domain have proven to be a successful tool in recent years. Of course, the smaller such a subregion, the tighter a polyhedral approximation can be made, but at the expense of considering many additional subregions. We propose to design polyhedral approximations by taking into account subdivisions of the value range, i.e., the image of the nonlinear function. This may yield a much lower number of subdivisions, if the value range is small. At the same time it serves to classify subdivisions of the domain by similar values, which helps branch-and-bound based solvers avoiding symmetric branches, or, equivalently, improves MILP-based reformulations. The value based decomposition can be formalized as an extended formulation of the convex hull of feasible points conv(x, f (x)). In the extended formulation, combinatorial cuts linking the new variables can be derived from the underlying nonlinear problem. We present a condition when the reformulation yields exact over-or underestimators, and examples of nonconvex functions that satisfy it. As an application, in a purely discrete setting, the extended formulation is explicitly characterized for the bilinear functional xy over rectangular subregions in all orthants.

In Proceedings of the 11th International Conference on Agents and Artificial Intelligence (ICAART 2019) - Volume 2, pages 294-305, 2019

A valued constraint satisfaction problem (VCSP) is a soft constraint framework that can formalize a wide range of applications related to Combinatorial Optimization and Artificial Intelligence. Most researchers have focused on the development of algorithms for solving mono-objective problems. However, many real-world satisfaction/optimization problems involve multiple objectives that should be considered separately and satisfied/optimized simultaneously. Solving a Multi-Objective Optimization Problem (MOP) consists of finding the set of all non-dominated solutions, known as the Pareto Front. In this paper, we introduce multi-objective valued constraint satisfaction problem (MO-VCSP), that is a VCSP involving multiple objectives, and we extend soft local arc consistency methods, which are widely used in solving Mono-Objective VCSP, in order to deal with the multi-objective case. Also, we present multi-objective enforcing algorithms of such soft local arc consistencies taking into account the Pareto principle. The new Pareto-based soft arc consistency (P-SAC) algorithms compute a Lower Bound Set of the efficient frontier. As a consequence, P-SAC can be integrated into a Multi-Objective Branch and Bound (MO-BnB) algorithm in order to ensure its pruning efficiency.