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

Abstract: We show that Set Cover on instances with $N$ elements cannot be approximated within $(1-\gamma)\ln N$-factor in time exp($N<sup>{\gamma-\delta})$, for any $0 &lt; \gamma &lt; 1$ and any $\delta &gt; 0$, 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 $exp(O(N<sup>\gamma))$. 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 $exp(\tilde{O}(N<sup>\gamma))$ time, for any $1/2 \le \gamma &lt; 1$.