Probabilistic analysis of an approximation algorithm for maximum subset sum using recurrence relations

Abstract

Given a positive integer M, and a set S = {x1, x2, ..., xn} of positive integers, the maximum subset sum problem is to find a subset S' of S such that Σxεs'^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 analyze the average-case performance of a simple on-line approximation algorithm assuming that all integers in S are independent and have the same probabilty 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 multiprogrammed parallel systems. The algorithm studied can also be easily adapted for dynamic job scheduling in such systems.