Maximizing k-Submodular Functions and Beyond

TL;DR

This paper studies the maximization of complex set functions called k-submodular functions, providing approximation algorithms with guarantees and establishing optimality bounds in the value oracle model.

Contribution

It introduces approximation algorithms for maximizing k-submodular functions and extends known results with new bounds and characterizations.

Findings

Deterministic greedy algorithm achieves 1/(1+r) approximation.

Randomized greedy algorithm achieves 1/(1+√(k/2)) approximation for r=k.

For k=r=2, the approximation guarantee is 1/2, proven to be optimal.

Abstract

We consider the maximization problem in the value oracle model of functions defined on $k$-tuples of sets that are submodular in every orthant and $r$-wise monotone, where $k\geq 2$ and $1\leq r\leq k$. We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of $1/(1+r)$. For $r=k$, we give an analysis of a randomised greedy algorithm that shows that any such function can be approximated to a factor of $1/(1+\sqrt{k/2})$. In the case of $k=r=2$, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of $1/2$. We show that, as in the case of submodular functions, this result is the best possible in both the value query model, and under the assumption that $NP\neq RP$. Extending a result of Ando et al., we show that for any $k\geq 3$ submodularity in every orthant and pairwise monotonicity (i.e. $r=2$) precisely characterize $k$-submodular functions. Consequently, we obtain an approximation guarantee of $1/3$ (and thus independent of $k$) for the maximization problem of $k$-submodular functions.