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We introduce a new class of functions that can be minimized in polynomialtime in the value oracle model. These are functions $f$ satisfying$f(x)+f(y)\ge f(x \sqcap y)+f(x \sqcup y)$ where the domain of each variable$x_i$ corresponds to nodes of a rooted binary tree, and operations$\sqcap,\sqcup$ are defined with respect to this tree. Special cases includepreviously studied $L^\natural$-convex and bisubmodular functions, which can beobtained with particular choices of trees. We present a polynomial-timealgorithm for minimizing functions in the new class. It combines Murota'ssteepest descent algorithm for $L^\natural$-convex functions with bisubmodularminimization algorithms.

Submodularity on a tree: Unifying $L^\natural$-convex and bisubmodular functions