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arXiv · 2008.05374

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

Marek Cygan·Magnús M. Halldórsson·Guy Kortsarz
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Abstract

We show that Set Cover on instances with $N$ elements cannot be approximated within $(1-γ)\ln N$-factor in time exp($N^{γ-δ})$, for any $0 < γ< 1$ and any $δ> 0$, assuming the Exponential Time Hypothesis. This essentially matches the best upper bound known by Cygan et al.\ (IPL, 2009) of $(1-γ)\ln N$-factor in time $exp(O(N^γ))$. 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 $(1-γ)\ln N$-approximation algorithm for these problems that runs in $exp(\tilde{O}(N^γ))$ time, for any $1/2 \le γ< 1$.

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Marek Cygan, Magnús M. Halldórsson, Guy Kortsarz. 2020-08-12. Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems. https://arxiv.org/abs/2008.05374

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