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

Vladimir Kolmogorov.

In 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), August 2011.


We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions $f$ satisfying $f(\bx)+f(\by)\ge f(\bx \sqcap \by)+f(\bx \sqcup \by)$ 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 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.


arXiv version