Average-case performance analysis of an approximation algorithm for maximum subset sum using recurrence relations

Given a positive integer M, and a set S = {x(1),x(2),...,x(n),) of positive integers, the maximum subset sum problem is to find a subset S' of S such that Sigma(x is an element of S') x is maximized under the constraint that the summation is no larger than M. In addition, the cardinality of S' is also a maximum among all subsets of S which achieve the maximum subset sum. This problem is known to be NP-hard. We analyse the average-case performance of a simple on-line approximation algorithm assuming that all integers in S are independent and have the same probability distribution. We develop a general methodology, i.e., using recurrence relations, to evaluate the expected values of the content and the cardinality of S' for any distribution. The maximum subset sum problem has important applications, especially in static job scheduling in multiprogammed parallel systems. The algorithm studied can also be easily adapted for dynamic job scheduling in such systems. (C) 1998 Elsevier Science Ltd. All rights reserved.