Abstract
AbstractWe introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions f satisfying \(f(\mbox{\boldmath $x$})+f(\mbox{\boldmath $y$})\ge f(\mbox{\boldmath $x$} \sqcap \mbox{\boldmath $y$})+f(\mbox{\boldmath $x$} \sqcup \mbox{\boldmath $y$})\) where the domain of each variable x i corresponds to nodes of a rooted binary tree, and operations ⊓ , ⊔ are defined with respect to this tree. Special cases include previously studied \(L^\natural\)-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 \(L^\natural\)-convex functions with bisubmodular minimization algorithms.KeywordsSubmodularity \(L^\natural\)-convexitybisubmodularityValued Constraint Satisfaction Problem (VCSP)
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