Submodular systems and related topics

Let Open image in new window be a distributive lattice formed by subsets of a finite set with set union and intersection as the lattice operations, and let f be a submodular function on Open image in new window . The pair ( Open image in new window , f) is called a submodular system and is a generalization of a (poly-)matroid. The present paper makes a survey of the author’s earlier work on submodular systems and provides a unifying view and some useful observations on related topics such as geometries on posets, generalized polymatroids, boundary hypermatroids, submodular functions on crossing families, submodular flows, strongly connected orientations of graphs, Lovasz’s extension of set functions, minimization of submodular functions etc. We also show a new approach to the problem of minimizing submodular functions.

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