Local Approximability of Max-Min and Min-Max Linear Programs

  • Authors:

  • Patrik Floréen;Marja Hassinen;Joel Kaasinen;Petteri Kaski;Topi Musto;Jukka Suomela

  • Affiliations:

  • University of Helsinki, Helsinki Institute for Information Technology HIIT, P.O. Box 68, 00014, Helsinki, Finland;University of Helsinki, Helsinki Institute for Information Technology HIIT, P.O. Box 68, 00014, Helsinki, Finland;University of Helsinki, Helsinki Institute for Information Technology HIIT, P.O. Box 68, 00014, Helsinki, Finland;University of Helsinki, Helsinki Institute for Information Technology HIIT, P.O. Box 68, 00014, Helsinki, Finland;University of Helsinki, Helsinki Institute for Information Technology HIIT, P.O. Box 68, 00014, Helsinki, Finland;University of Helsinki, Helsinki Institute for Information Technology HIIT, P.O. Box 68, 00014, Helsinki, Finland

  • Venue:
  • Theory of Computing Systems - Special Issue: Parallelism in Algorithms and Architectures
  • Year:
  • 2011

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Abstract

In a max-min LP, the objective is to maximise ω subject to A x≤1, C x≥ω 1, and x≥0. In a min-max LP, the objective is to minimise ρ subject to A x≤ρ 1, C x≥1, and x≥0. The matrices A and C are nonnegative and sparse: each row a i of A has at most ΔI positive elements, and each row c k of C has at most ΔK positive elements. We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any ΔI ≥2, ΔK ≥2, and ε0 there exists a local algorithm that achieves the approximation ratio ΔI (1−1/ΔK )+ε. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio ΔI (1−1/ΔK ) for any ΔI ≥2 and ΔK ≥2.