Maximizing Modular plus Non-monotone Submodular Functions

TL;DR

This paper introduces a new approximation algorithm for maximizing the sum of a non-monotone submodular and a modular function over down-closed sets, relaxing previous restrictions on the modular function's value range.

Contribution

It presents the first algorithm that handles arbitrary modular functions without positivity constraints, providing approximation guarantees based on the modular function's negative part ratio.

Findings

The algorithm achieves a guarantee close to 1/e for the combined function.

A hardness result shows no polynomial algorithm can surpass a 0.478 approximation ratio.

The work extends submodular maximization theory to more general modular functions.

Abstract

The research problem in this work is the relaxation of maximizing non-negative submodular plus modular with the entire real number domain as its value range over a family of down-closed sets. We seek a feasible point $\mathbf{x}^*$ in the polytope of the given constraint such that $\mathbf{x}^*\in\arg\max_{\mathbf{x}\in\mathcal{P}\subseteq[0,1]^n}F(\mathbf{x})+L(\mathbf{x})$, where $F$, $L$ denote the extensions of the underlying submodular function $f$ and modular function $\ell$. We provide an approximation algorithm named \textsc{Measured Continuous Greedy with Adaptive Weights}, which yields a guarantee $F(\mathbf{x})+L(\mathbf{x})\geq \left(1/e-\mathcal{O}(\epsilon)\right)\cdot f(OPT)+\left(\frac{\beta-e}{e(\beta-1)}-\mathcal{O}(\epsilon)\right)\cdot\ell(OPT)$ under the assumption that the ratio of non-negative part within $\ell(OPT)$ to the absolute value of its negative part is demonstrated by a parameter $\beta\in[0, \infty]$, where $OPT$ is the optimal integral solution for the discrete problem. It is obvious that the factor of $\ell(OPT)$ is $1$ when $\beta=0$, which means the negative part is completely dominant at this time; otherwise the factor is closed to $1/e$ whe $\beta\rightarrow\infty$. Our work first breaks the restriction on the specific value range of the modular function without assuming non-positivity or non-negativity as previous results and quantifies the relative variation of the approximation guarantee for optimal solutions with arbitrary structure. Moreover, we also give an analysis for the inapproximability of the problem we consider. We show a hardness result that there exists no polynomial algorithm whose output $S$ satisfies $f(S)+\ell(S)\geq0.478\cdot f(OPT)+\ell(OPT)$.