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Decreasing minimization on M-convex sets: background and structures


The present work is the first member of a pair of papers concerning decreasingly-minimal (dec-min) elements of a set of integral vectors, where a vector is dec-min if its largest component is as small as possible, within this, the next largest component is as small as possible, and so on. This discrete notion, along with its fractional counterpart, showed up earlier in the literature under various names. The domain we consider is an M-convex set, that is, the set of integral elements of an integral base-polyhedron. A fundamental difference between the fractional and the discrete case is that a base-polyhedron has always a unique dec-min element, while the set of dec-min elements of an M-convex set admits a rich structure, described here with the help of a ‘canonical chain’. As a consequence, we prove that this set arises from a matroid by translating the characteristic vectors of its bases with an integral vector. By relying on these characterizations, we prove that an element is dec-min if and only if the square-sum of its components is minimum, a property resulting in a new type of min-max theorems. The characterizations also give rise, as shown in the companion paper, to a strongly polynomial algorithm, and to several applications in the areas of resource allocation, network flow, matroid, and graph orientation problems, which actually provided a major motivation to the present investigations. In particular, we prove a conjecture on graph orientation.

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We are grateful to the six authors of the paper by Borradaile et al. [4] because that work triggered the present research (and this is so even if we realized later that there had been several related works). We thank S. Fujishige and S. Iwata for discussion about the history of convex minimization over base-polyhedra. We also thank A. Jüttner and T. Maehara for illuminating the essence of the Newton–Dinkelbach algorithm. J. Tapolcai kindly drew our attention to engineering applications in resource allocation. Z. Király played a similar role by finding an article which pointed to a work of Levin and Onn on decreasingly minimal optimization in matroid theory. We are also grateful to M. Kovács for drawing our attention to some important papers in the literature concerning fair resource allocation problems. Special thanks are due to T. Migler for her continuous availability to answer our questions concerning the paper [4] and the work by Borradaile, Migler, and Wilfong [5], which paper was also a prime driving force in our investigations. We are grateful to B. Shepherd and K. Bérczi for their advice that led to restructuring our presentation appropriately. We are also grateful to an anonymous referee of the paper whose strategic suggestions were particularly important to shape 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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The research was partially supported by the National Research, Development and Innovation Fund of Hungary (FK_18) – No. NKFI-128673. 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: background and structures. Math. Program. (2021).

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  • Submodular optimization
  • Matroid
  • Base-polyhedron
  • M-convex set
  • Lexicographic minimization

Mathematics Subject Classification

  • 90C27
  • 05C
  • 68R10