Local Search and Constraint Programming

LS techniques are based on a simple and general idea. Let P be the combinatorial optimization problem we want to solve, and s a current solution which, for the moment, we assume to be feasible for P , and to have value z(s). A neighborhood is defined for s with respect to a move type N , i.e., a function mapping s (actually, any feasible solution of P ) in a subset N (s) of the overall solution space. In other words, N (s) contains all the feasible solutions of P which can be reached from s by means of a move of type N . Examples of moves are given throughout the paper, but, roughly speaking, the move is a manipulation (and repair) of s whose effect is the transition to another solution x ∈ N (s). The LS framework explores the neighborhood by searching for the solution x∗ ∈ N (s) such that δz = z(s) − z(x∗) is maximized (for minimization problems). If δz > 0, then an improved solution has been found, and the process is iterated by considering x∗ as new current solution. Otherwise, a local optimum has been reached, and several very effective techniques can be applied to escape from it.