State Space Relaxation and Search Strategies in Dynamic Programming

Abstract

Few combinatorial optimization problems can be solved effectively by dynamic programming as the number of the vertices of the state space graph can be enormous.

State space relaxation, introduced by Christofides, Mingozzi and Toth [1] is a general relaxation method whereby the state-space graph of a given dynamic programming recursion is relaxed in such a way that the solution of the associated recursion in the relaxed state-space provides a lower bound to the original problem. This talk gives a survey of this methodology and gives, as examples, applications to some combinatorial optimization problems including the traveling salesman problem (TSP). We describe a more general relaxation method for dynamic programming which produces a ”reduced state space” containing a subset of the original states and the state space relaxation of the other states. Subgradient optimization and “state space decompression” are discussed as methods for improving the resulting lower bounds. We describe an iterative optimal search strategy in the original state space using bounding functions based on the reduced state space which explores, at each iteration, only a limited subset of states of the original state space graph. This procedure can also be used as a heuristic simply by limiting the maximum number of iterations. We give, as examples, applications to the TSP with time windows and precedence constraints and to the shortest path problem with additional constraints.