Submodularity on a tree: unifying L-convex and bisubmodular functions

Authors:
Vladimir Kolmogorov
Affiliations:
University College London, UK
Venue:
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Year:
2011

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Abstract

We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions f satisfying f(x) + f(y) ≥ f(x ∏ y) + f(x ∐ y) where the domain of each variable xi corresponds to nodes of a rooted binary tree, and operations ∏,∐ are defined with respect to this tree. Special cases include previously studied L-convex and bisubmodular functions, which can be obtained with particular choices of trees. We present a polynomial-time algorithm for minimizing functions in the new class. It combines Murota's steepest descent algorithm for L-convex functions with bisubmodular minimization algorithms.