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Improved Approximations for k-Exchange Systems

(Extended Abstract)
  • Moran Feldman
  • Joseph (Seffi) Naor
  • Roy Schwartz
  • Justin Ward
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)

Abstract

Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k + 1)-approximation for maximizing a monotone submodular function over a k-system. For the special case of k-matroid intersection, a local search approach was recently shown to provide an improved approximation of 1 / (k + δ) for arbitrary δ > 0. Unfortunately, many fundamental optimization problems are represented by a k-system which is not a k-intersection. An interesting question is whether the local search approach can be extended to include such problems.

We answer this question affirmatively. Motivated by the b-matching and k-set packing problems, as well as the more general matroid k-parity problem, we introduce a new class of set systems called k-exchange systems, that includes k-set packing, b-matching, matroid k-parity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k + 1)-claw free graphs, asymmetric TSP, job interval selection with identical lengths and frequency allocation on lines. We give a natural local search algorithm which improves upon the current greedy approximation, for this new class of independence systems. Unlike known local search algorithms for similar problems, we use counting arguments to bound the performance of our algorithm.

Moreover, we consider additional objective functions and provide improved approximations for them as well. In the case of linear objective functions, we give a non-oblivious local search algorithm, that improves upon existing local search approaches for matroid k-parity.

Keywords

Local Search Local Search Algorithm Free Graph Submodular Function Linear Objective Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Moran Feldman
    • 1
  • Joseph (Seffi) Naor
    • 1
  • Roy Schwartz
    • 1
  • Justin Ward
    • 2
  1. 1.Computer Science Dept.TechnionHaifaIsrael
  2. 2.Computer Science Dept.University of TorontoTorontoCanada

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