Integral Simplex Method for Solving Combinatorial Optimization Problems

摘要

In this paper a local integral simplex method will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex method creates and solves a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc., and the solution to at least one of which is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is the optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n. Preliminary computational experience is given which indicates that global method has a low order polynomial empirical performance when solving such problems.