Fast algorithms for k-submodular maximization subject to a matroid constraint

TL;DR

This paper introduces a Threshold-Decreasing Algorithm for maximizing k-submodular functions under matroid constraints, achieving improved query complexity with near-optimal approximation ratios for both monotone and non-monotone cases.

Contribution

It presents a novel threshold-decreasing algorithm that reduces query complexity for k-submodular maximization under matroid constraints, with new approximation guarantees.

Findings

Achieves a (1/2 - ε)-approximation for monotone k-submodular maximization.

Achieves a (1/3 - ε)-approximation for non-monotone case.

Reduces query complexity compared to greedy algorithms.

Abstract

In this paper, we apply a Threshold-Decreasing Algorithm to maximize $k$-submodular functions under a matroid constraint, which reduces the query complexity of the algorithm compared to the greedy algorithm with little loss in approximation ratio. We give a $(\frac{1}{2} - \epsilon)$-approximation algorithm for monotone $k$-submodular function maximization, and a $(\frac{1}{3} - \epsilon)$-approximation algorithm for non-monotone case, with complexity $O(\frac{n(k\cdot EO + IO)}{\epsilon} \log \frac{r}{\epsilon})$, where $r$ denotes the rank of the matroid, and $IO, EO$ denote the number of oracles to evaluate whether a subset is an independent set and to compute the function value of $f$, respectively. Since the constraint of total size can be looked as a special matroid, called uniform matroid, then we present the fast algorithm for maximizing $k$-submodular functions subject to a total size constraint as corollaries. corollaries.