Improved Inapproximability For Submodular Maximization

TL;DR

This paper establishes tighter hardness bounds for approximating the maximum of submodular and symmetric submodular functions, showing these problems are harder to approximate than previously known, under the Unique Games conjecture.

Contribution

It improves the inapproximability bounds for submodular maximization problems, refining the understanding of their computational hardness.

Findings

Approximation within 0.695 for general submodular maximization is Unique Games-hard.

Approximation within 0.739 for symmetric submodular maximization is Unique Games-hard.

These bounds are tighter than previous NP-hardness results.

Abstract

We show that it is Unique Games-hard to approximate the maximum of a submodular function to within a factor 0.695, and that it is Unique Games-hard to approximate the maximum of a symmetric submodular function to within a factor 0.739. These results slightly improve previous results by Feige, Mirrokni and Vondr\'ak (FOCS 2007) who showed that these problems are NP-hard to approximate to within $3/4 + \epsilon \approx 0.750$ and $5/6 + \epsilon \approx 0.833$, respectively.