Solving Constraint Satisfaction Research Articles

Quantum-accelerated constraint programming

Constraint programming (CP) is a paradigm used to model and solve constraint satisfaction and combinatorial optimization problems. In CP, problems are modeled with constraints that describe acceptable solutions and solved with backtracking tree search augmented with logical inference. In this paper, we show how quantum algorithms can accelerate CP, at both the levels of inference and search. Leveraging existing quantum algorithms, we introduce a quantum-accelerated filtering algorithm for thealldifferentglobal constraint and discuss its applicability to a broader family of global constraints with similar structure. We propose frameworks for the integration of quantum filtering algorithms within both classical and quantum backtracking search schemes, including a novel hybrid classical-quantum backtracking search method. This work suggests that CP is a promising candidate application for early fault-tolerant quantum computers and beyond.

A Hybrid Programming Framework for Modeling and Solving Constraint Satisfaction and Optimization Problems

This paper proposes a hybrid programming framework for modeling and solving of constraint satisfaction problems (CSPs) and constraint optimization problems (COPs). Two paradigms, CLP (constraint logic programming) and MP (mathematical programming), are integrated in the framework. The integration is supplemented with the original method of problem transformation, used in the framework as a presolving method. The transformation substantially reduces the feasible solution space. The framework automatically generates CSP and COP models based on current values of data instances, questions asked by a user, and set of predicates and facts of the problem being modeled, which altogether constitute a knowledge database for the given problem. This dynamic generation of dedicated models, based on the knowledge base, together with the parameters changing externally, for example, the user’s questions, is the implementation of the autonomous search concept. The models are solved using the internal or external solvers integrated with the framework. The architecture of the framework as well as its implementation outline is also included in the paper. The effectiveness of the framework regarding the modeling and solution search is assessed through the illustrative examples relating to scheduling problems with additional constrained resources.

Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming

Message-passing algorithms based on belief-propagation (BP) are successfully used in many applications, including decoding error correcting codes and solving constraint satisfaction and inference problems. The BP-based algorithms operate over graph representations, called factor graphs, that are used to model the input. Although in many cases, the BP-based algorithms exhibit impressive empirical results, not much has been proved when the factor graphs have cycles. This paper deals with packing and covering integer programs in which the constraint matrix is zero-one, the constraint vector is integral, and the variables are subject to box constraints. We study the performance of the min-sum algorithm when applied to the corresponding factor graph models of packing and covering linear programmings (LPs). We compare the solutions computed by the min-sum algorithm for packing and covering problems to the optimal solutions of the corresponding LP relaxations. In particular, we prove that if the LP has an optimal fractional solution, then for each fractional component, the min-sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction. This implies that the min-sum algorithm computes the optimal integral solution only if the LP has a unique optimal solution that is integral. The converse is not true in general. For a special case of packing and covering problems, we prove that if the LP has a unique optimal solution that is integral and on the boundary of the box constraints, then the min-sum algorithm computes the optimal solution in pseudopolynomial time. Our results unify and extend recent results for the maximum weight matching problem and for the maximum weight independent set problem.

Solving constraint satisfaction and optimization problems by a neuro-fuzzy approach

The solution of constrained satisfaction and constrained optimization problems using a Hopfield model requires determination of the values of a certain number of coefficients linked to the surrounding conditions of the problem. It is quite difficult to determine these values, mainly because a heuristic search is necessary. This is not only time-consuming but may lead to solutions that are far from optimal, or even nonvalid ones. So far, there have been no works in literature offering a general method for the search for coefficents with will guarantee optimal or close to optimal solutions. This paper proposes a fuzzy approach which allows automatic determination of Hopfield coefficients.