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Proximity theorems of discrete convex functions

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Abstract.

A proximity theorem is a statement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in a certain neighborhood of a solution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for L-convex and M-convex functions, this paper establishes proximity theorems for larger classes of discrete convex functions, L2-convex functions and M2-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem.

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan; and PRESTO, JST, Tokyo, Japan

    Kazuo Murota

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Akihisa Tamura

Authors
  1. Kazuo Murota
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  2. Akihisa Tamura
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Correspondence to Akihisa Tamura.

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Mathematics Subject Classification (2000): 90C27, 05B35

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Murota, K., Tamura, A. Proximity theorems of discrete convex functions. Math. Program., Ser. A 99, 539–562 (2004). https://doi.org/10.1007/s10107-003-0466-7

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  • DOI: https://doi.org/10.1007/s10107-003-0466-7

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