Submodularity on a tree: Unifying $L^{\mathrm{♮}}$-convex and bisubmodular functions

We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions $f$ satisfying $f(x)+f(y)\ge f(x\sqcap y)+f(x\bigsqcup y)$ where the domain of each variable $x_i$ corresponds to nodes of a rooted binary tree, and operations $\sqcap ,\bigsqcup$ are defined with respect to this tree. Special cases include previously studied $L^{\mathrm{♮}}$-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 $L^{\mathrm{♮}}$-convex functions with bisubmodular minimization algorithms.