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arXiv · 1903.08343

Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an ${\rm M}^{\natural}$-Concave Set Function

Tomohito Fujii·Shuji Kijima
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Abstract

Submodularity is an important concept in combinatorial optimization, and it is often regarded as a discrete analog of convexity. It is a fundamental fact that the set of minimizers of any submodular function forms a distributive lattice. Conversely, it is also known that any finite distributive lattice is isomorphic to the minimizer set of a submodular function, through the celebrated Birkhoff's representation theorem. ${\rm M}^{\natural}$-concavity is a key concept in discrete convex analysis. It is known for set functions that the class of ${\rm M}^{\natural}$-concavity is a proper subclass of submodularity. Thus, the minimizer set of an ${\rm M}^{\natural}$-concave function forms a distributive lattice. It is natural to ask if any finite distributive lattice appears as the minimizer set of an ${\rm M}^{\natural}$-concave function. This paper affirmatively answers the question.

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Tomohito Fujii, Shuji Kijima. 2019-03-20. Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an ${\rm M}^{\natural}$-Concave Set Function. https://arxiv.org/abs/1903.08343

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