Approximation algorithms for $k$-submodular maximization subject to a knapsack constraint

TL;DR

This paper develops approximation algorithms for maximizing $k$-submodular functions under a knapsack constraint, providing new algorithms with proven approximation ratios for both monotone and non-monotone cases.

Contribution

It introduces the first greedy algorithms with approximation guarantees for $k$-submodular maximization under knapsack constraints, covering both monotone and non-monotone functions.

Findings

Monotone case approximation ratio: ~0.432

Non-monotone case approximation ratio: ~0.317

First to address knapsack constraints for non-monotone $k$-submodular functions

Abstract

In this paper, we study the problem of maximizing $k$-submodular functions subject to a knapsack constraint. For monotone objective functions, we present a $\frac{1}{2}(1-e^{-2})\approx 0.432$ greedy approximation algorithm. For the non-monotone case, we are the first to consider the knapsack problem and provide a greedy-type combinatorial algorithm with approximation ratio $\frac{1}{3}(1-e^{-3})\approx 0.317$.