Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems

TL;DR

This paper establishes tight exponential time lower bounds for approximating Set Cover within certain factors, assuming ETH, and extends these bounds to related problems, while also providing improved algorithms for some cases.

Contribution

It provides the first tight exponential time lower bounds for Set Cover approximation under ETH and applies similar bounds to related problems, along with new algorithms.

Findings

Set Cover cannot be approximated within (1-γ)ln N in exp(N^{γ-δ}) time assuming ETH.

Matching upper bounds are confirmed by prior work, showing tightness.

New approximation algorithms are developed for related problems with specific runtime guarantees.

Abstract

We show that Set Cover on instances with $N$ elements cannot be approximated within $(1-\gamma)\ln N$-factor in time exp($N^{\gamma-\delta})$, for any $0 < \gamma < 1$ and any $\delta > 0$, assuming the Exponential Time Hypothesis. This essentially matches the best upper bound known by Cygan et al.\ (IPL, 2009) of $(1-\gamma)\ln N$-factor in time $exp(O(N^\gamma))$. The lower bound is obtained by extracting a standalone reduction from Label Cover to Set Cover from the work of Moshkovitz (Theory of Computing, 2015), and applying it to a different PCP theorem than done there. We also obtain a tighter lower bound when conditioning on the Projection Games Conjecture. We also treat three problems (Directed Steiner Tree, Submodular Cover, and Connected Polymatroid) that strictly generalize Set Cover. We give a $(1-\gamma)\ln N$-approximation algorithm for these problems that runs in $exp(\tilde{O}(N^\gamma))$ time, for any $1/2 \le \gamma < 1$.