An efficient fully polynomial approximation scheme for the Subset-Sum Problem

• Published in: Journal of computer and system sciences Vol. 66; no. 2; pp. 349 - 370
• Main Authors: Kellerer, Hans, Mansini, Renata, Pferschy, Ulrich, Speranza, Maria Grazia
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2003
• Subjects: Fully polynomial approximation scheme; Knapsack problem; Subset-sum problem; Worst-case performance; Fully polynomial approximation scheme; Subset-sum problem; Worst-case performance; Knapsack problem
• ISSN: 0022-0000; 1090-2724
• DOI: 10.1016/S0022-0000(03)00006-0

Given a set of n positive integers and a knapsack of capacity c, the Subset-Sum Problem is to find a subset the sum of which is closest to c without exceeding the value c. In this paper we present a fully polynomial approximation scheme which solves the Subset-Sum Problem with accuracy ε in time O( min{n·1/ε,n+1/ε 2 log(1/ε)}) and space O( n+1/ ε). This scheme has a better time and space complexity than previously known approximation schemes. Moreover, the scheme always finds the optimal solution if it is smaller than (1− ε) c. Computational results show that the scheme efficiently solves instances with up to 5000 items with a guaranteed relative error smaller than 1/1000.