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

  • Authors:

  • Hans Kellerer;Renata Mansini;Ulrich Pferschy;Maria Grazia Speranza

  • Affiliations:

  • Institut für Statistik und Operations Research, Universität Graz, Universitätsstr. 15, A-8010 Graz, Austria;Dipartimento di Elettronica per l'Automazione, Università di Brescia, via Branze 38, I-25123 Brescia, Italy;Institut für Statistik und Operations Research, Universität Graz, Universitätsstr. 15, A-8010 Graz, Austria;Dipartimento Metodi Quantitativi, Università di Brescia, Contrada S. Chiara 48/b, I-25122 Brescia, Italy

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2003

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Abstract

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/ε2 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.