Time-approximation trade-offs for inapproximable problems

Édouard Bonnet, Michael Lampis, Vangelis Th. Paschos
2018 Journal of computer and system sciences (Print)  
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 r n/r . We show that, for most values of r, if this running
more » ... ime could be significantly improved the ETH would fail. For Max Minimal Vertex Cover we give a nontrivial √ r-approximation in time 2 n/r . We match this with a similarly tight result. We also give a log r-approximation for Min ATSP in time 2 n/r and an r-approximation for Max Grundy Coloring in time r n/r . Furthermore, we show that Min Set Cover exhibits a curious behavior in this superpolynomial setting: for any δ > 0 it admits an m δ -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. 21 Dana Moshkovitz. The projection games conjecture and the NP-hardness of ln napproximating set-cover. In A.
doi:10.1016/j.jcss.2017.09.009 fatcat:owftfddbcvejvlsb72epihnxfq