Neighborhood search algorithms for finding optimal traveling salesman tours must be inefficient

Abstract

search In this paper, we explore the use of neighborhood search techniques for finding optimal solutions to the symmetric Traveling Salesman Problem. These techniques have been dramatically successful in obtaining near-optimal solutions to this problem for a reasonable expenditure of effort (1,2,3,4,5,6,9,10,12). Extensions of these techniques can be used to obtain the globally optimum solution, but the effort involved is at least an exponential function of the number of cities, n. Indeed, as this paper demonstrates, all local search algorithms that are capable of finding the optimal solution to an arbitrary n-city problem must grow at least as fast as equation !. Thus for large problems, these algorithms are computationally inefficient. In following section we show that any exact neighborhood search algorithm for the Traveling Salesman Problem must inspect a prohibitively large number of feasible solutions. We begin with a brief discussion of the Traveling Salesman Problem (TSP) and neighborhood search techniques in section II. In section III we develop a necessary condition for neighborhood search to converge to an optimal solution. We use this result in sections IV and V to obtain a lower bound on the effectiveness of neighborhood search as applied to the TSP.