An Approximate Solution Method for Discrete Constraint Optimization Problem Using Linear Programming

Abstract

Many problems in AI can be formulated as Constraint Satisfaction Problems (CSPs). In actual problems, Constraint Optimization Problem (COP) is often more important. In this paber, we present a new efficient method for COP, more exactly, for VCOP (value-based COP), in which every value of CSP is assigned an integer cost and the CSP solution with a minimal cost is to be found. Since the computational cost of COP is very high, i.e., NP-hard, simple search methods working in binary (0-1) space are not good enough in terms of efficiency. Thus we try to utilize a linear programming method, namely simplex method, to find a near-optimal solution efliciently. In our method, COP is transformed at first into a 0-1 integer linear programming problem. Then we solve it's relaxation problem which allows a real-number solution rather than a O-1 solution. With using the real-number optimal solution, we execute a local search method called GLS (guided local search) to find a 0-l near-optimal solution of COP. GLS is a general and compact optimization technique with a capability of escaping from local optimal points. In our method, the initial search point for GLS is determined referring to the real-number optimal solution of the simplex method. Also, terms regarding the distance between a temporal search point and the real-number optimal solution are added into the augmented cost function of GLS to control the search to intensively investigate the neighborhood of the real-number optimal solution. This method shows a good performance empirically in computational efEiciency and in solution quality on COP, especially on loosely constrained COP. In a wide sense, our research here has shown that the combination of the linear programming technique with search method is beneficial to achieve search efficiency in constraint optimization.