Tight Local Approximation Results for Max-Min Linear Programs

  • Authors:

  • Patrik Floréen;Marja Hassinen;Petteri Kaski;Jukka Suomela

  • Affiliations:

  • Helsinki Institute for Information Technology HIIT, Helsinki University of Technology and University of Helsinki, University of Helsinki, Finland FI-00014;Helsinki Institute for Information Technology HIIT, Helsinki University of Technology and University of Helsinki, University of Helsinki, Finland FI-00014;Helsinki Institute for Information Technology HIIT, Helsinki University of Technology and University of Helsinki, University of Helsinki, Finland FI-00014;Helsinki Institute for Information Technology HIIT, Helsinki University of Technology and University of Helsinki, University of Helsinki, Finland FI-00014

  • Venue:
  • Algorithmic Aspects of Wireless Sensor Networks
  • Year:
  • 2008

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Abstract

In a bipartite max-min LP, we are given a bipartite graph$\mathcal{G} = (V \cup I \cup K, E)$, where each agentv ∈ V is adjacent to exactly one constrainti ∈ I and exactly one objectivek ∈ K. Each agent v controls a variable x v .For each i ∈ I we have a nonnegative linearconstraint on the variables of adjacent agents. For eachk ∈ K we have a nonnegative linear objectivefunction of the variables of adjacent agents. The task is tomaximise the minimum of the objective functions. We study localalgorithms where each agent v must choose x v based on input withinits constant-radius neighbourhood in . We show that for everyε 0 there exists a local algorithm achieving theapproximation ratio Δ I (1 − 1/Δ K) + ε. We also show that this result is thebest possible – no local algorithm can achieve theapproximation ratio Δ I (1 − 1/Δ K). Here Δ I is the maximum degree of a vertexi ∈ I, and Δ K is the maximum degree of avertex k ∈ K. As a methodological contribution,we introduce the technique of graph unfolding for the design oflocal approximation algorithms.