Improved Approximation Algorithms for k-Submodular Function Maximization

TL;DR

This paper introduces improved polynomial-time approximation algorithms for maximizing nonnegative and monotone k-submodular functions, achieving better bounds and establishing tight hardness results, and extends methods to skew-bisubmodular functions.

Contribution

It presents a new 1/2-approximation algorithm for nonnegative k-submodular maximization, improves approximation ratios for monotone cases, and proves exponential query complexity lower bounds, extending to skew-bisubmodular functions.

Findings

New 1/2-approximation algorithm for nonnegative k-submodular functions

Improved approximation ratio of k/(2k-1) for monotone k-submodular functions

Hardness results showing exponential query complexity for certain approximations

Abstract

This paper presents a polynomial-time $1/2$-approximation algorithm for maximizing nonnegative $k$-submodular functions. This improves upon the previous $\max\{1/3, 1/(1+a)\}$-approximation by Ward and \v{Z}ivn\'y~(SODA'14), where $a=\max\{1, \sqrt{(k-1)/4}\}$. We also show that for monotone $k$-submodular functions there is a polynomial-time $k/(2k-1)$-approximation algorithm while for any $\varepsilon>0$ a $((k+1)/2k+\varepsilon)$-approximation algorithm for maximizing monotone $k$-submodular functions would require exponentially many queries. In particular, our hardness result implies that our algorithms are asymptotically tight. We also extend the approach to provide constant factor approximation algorithms for maximizing skew-bisubmodular functions, which were recently introduced as generalizations of bisubmodular functions.