Submodularity on a Tree: Unifying $L^\natural$ -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) ≥ 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.