Time-Approximation Trade-offs for Inapproximable Problems

In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set, Max Induced Path, Forest and Tree, for any r(n), a simple, known scheme gives an approximation ratio of r in time roughly rn/r. We show that, for most values of r, if this running time could be significantly improved the ETH would fail. For Max Minimal Vertex Cover we give a nontrivial √r-approximation in time 2n/r. We match this with a similarly tight result. We also give a log r-approximation for Min ATSP in time 2n/r and an r-approximation for Max Grundy Coloring in time rn/r. Furthermore, we show that Min Set Cover exhibits a curious behavior in this superpolynomial setting: for any δ > 0 it admits an mδ-approximation, where m is the number of sets, in just quasi-polynomial time. We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail. © Édouard Bonnet, Michael Lampis and Vangelis Th. Paschos; licensed under Creative Commons License CC-BY.