
On subexponential running times for approximating directed Steiner tree and related problems
This paper concerns proving almost tight (superpolynomial) running time...
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Set Cover in Sublinear Time
We study the classic set cover problem from the perspective of sublinea...
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Revisiting the Set Cover Conjecture
In the Set Cover problem, the input is a ground set of n elements and a ...
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A Chronology of Set Cover Inapproximability Results
It is wellknown that an algorithm exists which approximates the NPcomp...
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A Tight Bound for Stochastic Submodular Cover
We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) ...
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Nearly Optimal Time Bounds for kPath in Hypergraphs
In the kPath problem the input is a graph G and an integer k, and the go...
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The Batched Set Cover Problem
We introduce the batched set cover problem, which is a generalization of...
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Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems
We show that Set Cover on instances with N elements cannot be approximated within (1γ)ln Nfactor in time exp(N^γδ), 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 (1γ)ln Nfactor 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 (1γ)ln Napproximation algorithm for these problems that runs in exp(Õ(N^γ)) time, for any 1/2 ≤γ < 1.
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