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Decreasing minimization on M-convex sets: algorithms and applications


This paper is concerned with algorithms and applications of decreasing minimization on an M-convex set, which is the set of integral elements of an integral base-polyhedron. Based on a recent characterization of decreasingly minimal (dec-min) elements, we develop a strongly polynomial algorithm for computing a dec-min element of an M-convex set. The matroidal feature of the set of dec-min elements makes it possible to compute a minimum cost dec-min element, as well. Our second goal is to exhibit various applications in matroid and network optimization, resource allocation, and (hyper)graph orientation. We extend earlier results on semi-matchings to a large degree by developing a structural description of dec-min in-degree bounded orientations of a graph. This characterization gives rise to a strongly polynomial algorithm for finding a minimum edge-cost dec-min orientation.

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We are particularly grateful to an anonymous referee of the paper whose fundamental suggestions made a significant contribution to the final form of our work. This research was supported through the program “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2019. The two weeks we could spend at Oberwolfach provided an exceptional opportunity to conduct particularly intensive research. The research was partially supported by the National Research, Development and Innovation Fund of Hungary (FK_18) – No. NKFI-128673, and by CREST, JST, Grant Number JPMJCR14D2, Japan, and JSPS KAKENHI Grant Numbers JP26280004, JP20K11697.

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Correspondence to Kazuo Murota.

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A. Frank: The research was partially supported by the National Research, Development and Innovation Fund of Hungary (FK_18) – No. NKFI-128673. K. Murota: The research was supported by CREST, JST, Grant Number JPMJCR14D2, Japan, and JSPS KAKENHI Grant Numbers JP26280004, JP20K11697.

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Frank, A., Murota, K. Decreasing minimization on M-convex sets: algorithms and applications. Math. Program. (2021).

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  • Network flows
  • Resource allocation
  • Graph orientation
  • Decreasing minimization
  • M-convex set
  • Polynomial algorithm

Mathematics Subject Classification

  • 90C27
  • 05C
  • 68R10