On the approximability of the maximum feasible subsystem problem with 0/1-coefficients

  • Authors:

  • Khaled Elbassioni;Rajiv Raman;Saurabh Ray;René Sitters

  • Affiliations:

  • Max-Planck-Institut für Informatik, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany;Universität des Saarlandes, Saarbrücken, Germany;Eindhoven University of Technology, the Netherlands

  • Venue:
  • SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
  • Year:
  • 2009

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Abstract

Given a system of constraints li ≤ aTix ≤ ui, where ai ∈ {0, 1}n, and li, ui ∈ R+, for i = 1,..., m, we consider the problem Mrfs of finding the largest subsystem for which there exists a feasible solution x ≥ 0. We present approximation algorithms and inapproximability results for this problem, and study some important special cases. Our main contributions are: 1. In the general case, where ai ε {0, 1}n, a sharp separation in the approximability between the case when L = max{l1, ..., lm} is bounded above by a polynomial in n and m, and the case when it is not. 2. In the case where A is an interval matrix, a sharp separation in approximability between the case where we allow a violation of the upper bounds by at most a (1 + ε) factor, for any fixed ε 0 and the case where no violations are allowed. Along the way, we prove that the induced matching problem on bipartite graphs is inapproximable beyond a factor of Ω(n1/3-ε), for any ε 0 unless NP=ZPP. Finally, we also show applications of Mrfs to some recently studied pricing problems.