Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an M^-Concave Set Function

03/20/2019

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by   Tomohito Fujii, et al.

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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. M^-concavity is a key concept in discrete convex analysis. It is known for set functions that the class of M^-concave is a proper subclass of submodular. Thus, the minimizer set of an M^-concave function forms a distributive lattice. It is natural to ask if any finite distributive lattice appears as the minimizer set of an M^-concave function. This paper affirmatively answers the question.