A compact representation for minimizers of $k$-submodular functions

TL;DR

This paper introduces a compact poset-based representation for minimizers of $k$-submodular functions, generalizing previous models, with algorithms for construction and enumeration applicable to computer vision tasks.

Contribution

It generalizes the signed poset representation to $k$-submodular functions and provides algorithms for constructing and enumerating minimizers in various scenarios.

Findings

Complete characterization of elementary PIPs for $k$-submodular functions.

Algorithms for constructing PIPs from oracles, network-representability, and Potts models.

Efficient enumeration of maximal minimizers for Potts $k$-submodular functions.

Abstract

A $k$-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a $k$-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando-Fujishige's signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from $k$-submodular functions. We give algorithms to construct the elementary PIP of minimizers of a $k$-submodular function $f$ for three cases: (i) a minimizing oracle of $f$ is available, (ii) $f$ is network-representable, and (iii) $f$ arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts $k$-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.