A survey of very large-scale neighborhood search techniques

Many optimization problems of practical interest are computationally intractable. Therefore, a practical approach for solving such problems is to employ heuristic (approximation) algorithms that can find nearly optimal solutions within a reasonable amount of computation time. An improvement algorithm is a heuristic algorithm that generally starts with a feasible solution and iteratively tries to obtain a better solution. Neighborhood search algorithms (alternatively called local search algorithms) are a wide class of improvement algorithms where at each iteration an improving solution is found by searching the "neighborhood" of the current solution. A critical issue in the design of a neighborhood search algorithm is the choice of the neighborhood structure, that is, the manner in which the neighborhood is defined. As a rule of thumb, the larger the neighborhood, the better is the quality of the locally optimal solutions, and the greater is the accuracy of the final solution that is obtained. At the same time, the larger the neighborhood, the longer it takes to search the neighborhood at each iteration. For this reason, a larger neighborhood does not necessarily produce a more effective heuristic unless one can search the larger neighborhood in a very efficient manner. This paper concentrates on neighborhood search algorithms where the size of the neighborhood is "very large" with respect to the size of the input data and in which the neighborhood is searched in an efficient manner. We survey three broad classes of very large-scale neighborhood search (VLSN) algorithms: (1) variable-depth methods in which large neighborhoods are searched heuristically, (2) large neighborhoods in which the neighborhoods are searched using network flow techniques or dynamic programming, and (3) large neighborhoods induced by restrictions of the original problem that are solvable in polynomial time.