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Submodular Cost Allocation Problem and Applications

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Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6755))

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Abstract

We study the Minimum Submodular-Cost Allocation problem (MSCA). In this problem we are given a finite ground set V and k non-negative submodular set functions f 1,…,f k on V. The objective is to partition V into k (possibly empty) sets A 1, ⋯ , A k such that the sum ∑  i = 1 k f i (A i ) is minimized. Several well-studied problems such as the non-metric facility location problem, multiway-cut in graphs and hypergraphs, and uniform metric labeling and its generalizations can be shown to be special cases of MSCA. In this paper we consider a convex-programming relaxation obtained via the Lovász-extension for submodular functions. This allows us to understand several previous relaxations and rounding procedures in a unified fashion and also develop new formulations and approximation algorithms for related problems. In particular, we give a (1.5 − 1/k)-approximation for the hypergraph multiway partition problem. We also give a min {2(1 − 1/k), H Δ}-approximation for the hypergraph multiway cut problem when Δ is the maximum hyperedge size. Both problems generalize the multiway cut problem in graphs and the hypergraph cut problem is approximation equivalent to the node-weighted multiway cut problem in graphs.

This is an extended abstract without proofs. A longer version of the paper will be made available on the arXiv. The authors are supported in part by United States NSF grants CCF-0728782 and CCF-1016684.

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Authors and Affiliations

  1. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Chandra Chekuri & Alina Ene

Authors
  1. Chandra Chekuri
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  2. Alina Ene
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Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

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Chekuri, C., Ene, A. (2011). Submodular Cost Allocation Problem and Applications. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_30

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  • DOI: https://doi.org/10.1007/978-3-642-22006-7_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22005-0

  • Online ISBN: 978-3-642-22006-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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