Max-Min Optimization of the Multiple Knapsack Problem: an Implicit Enumeration Approach

Abstract

The binary knapsack problem is fundamental in combinatorial optimization, and algorithms to solve problems with nearly a million items are now available through the Internet. This paper is concerned with a variation of the problem, where there are n items to be packed into m knapsacks. Our problem is to find the assignment of items into knapsacks such that the minimum of the knapsack profits is maximized. This problem is referred to as the max-min multiple knapsack problem (M ³ KP). First, some upper bounds and a heuristic algorithm are presented, and based on these, we explore algorithms to solve the problem to optimally. Then, we make use of a novel pruning method to develop an implicit enumeration algorithm that can solve M ³ KPs with up to a few hundred items exactly.