Algorithms for Some Hard Knapsack Problems

Abstract

Knapsack problems are typically concerned with selecting from a set of n given items, each with a specified weight and value, a subset of items whose weight sum does not exceed a prescribed capacity and whose value is maximum. This NP-hard problem arises in many applications and has been the focus of considerable research over the past two decades. A number of exact algorithms have been developed for the classical 0–1 Knapsack Problems and its variants. In this paper, exact algorithms are presented for the following variants and data types: the Subset Sum Problem; the Strongly Correlated 0–1 Knapsack Problem; the Inverse Strongly Correlated 0–1 Knapsack Problem; and the corresponding Bounded Strongly Correlated Knapsack Problem and Bounded Subset Sum Problem. All our algorithms consist of three stages: the first stage generates an initial solution by a greedy procedure; the second stage refines the approximate solution; and the final stage applies a partial lexicographic search procedure to generate an optimal solution. Extensive computational experiments show that our algorithms are able to solve large problems of size up to one million variables in less than 7 seconds CPU time on a Silicon Graphic Workstation (R 5000) running at a clock speed of 150 MHz. A comparative analysis with some recent effective algorithms is given.