SCLP Semantics for (multi-criteria) Shortest Path Problems

The aim of the paper is to show an example of interaction between the AI and OR elds of research using the Constraint Programming (CP) framework to glue them together. We solve a shortest path problem using the c-semiring-based framework for programming over soft constraints (SCLP), which is an extension of classical constraint solving and programming. The SCLP approac handle uniformly several semirings.tion. Moreover, the cartesian product of two c-semirings is still a c-semiring. Using SCLP, a variety of multi-criteria optimization can be represented and solved in a uniform way, and heuristics and local search algorithms already used in the OR community for a totally ordered criterion, may be studied and extended to partial orders of (one or more) criteria. 1 Motivations Shortest Path (SP) problems 4] are among the most studied network optimization problems. They are mainly used to represent and solve transportation problems, where the optimization may involve diierent criteria, says cost, time, resources, etc. Most interesting is the multi-criteria case, where the optimization involves a set of criteria and the solution algorithms are extensions of the ordinary ones 4]. In this paper we consider multi-criteria SP problems and we suggest to model them via constraint logic programs (CLP) over soft constraints. The basic idea is that soft constraints allow to faithfully represent the optimization criteria, and CLP provides a declarative way to describe the SP problems. Moreover, this way of modelling and solving SP problems allows to associate to such problems both a declarative and an operational semantics. The paper is organized as follows: Section 2 gives the basic notions of soft constraints (using the semiring-based framework) and soft CLP. Then, Section 3 shows the construction to pass from an SP problem to a CLP program, and nally Section 4 concludes the paper by summarizing its contributions. 2 Soft CLP In this paper we will use the semiring-based CLP framework (SCLP) 3] to describe SP problems in a declarative fashion. The framework is based on the notion of c-semiring introduced in 1, 2]. A c-semiring S is a tuple hA; +; ; 0; 1i where A is a set with two special elements (0; 1 2 A) and with two operations + and that satisfy certain properties: + is closed, commutative, associative, idempotent, 0 is its unit element and 1 is its absorbing element; is closed, associative, commutative, distributes over +, 1 is its unit element, and …