Time-Approximation Trade-offs for Inapproximable Problems

TL;DR

This paper investigates how the approximability of certain inapproximable problems improves with increased computational time, establishing tight bounds and trade-offs that relate to ETH and other complexity conjectures.

Contribution

It introduces new time-approximation trade-offs for multiple problems, showing tight bounds and connecting these to ETH and other complexity assumptions.

Findings

Approximation ratios improve with increased runtime for various problems.

Most improvements would contradict ETH, indicating tight bounds.

Min Set Cover exhibits rapid approximation improvements in quasi-polynomial time.

Abstract

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 time could be significantly improved the ETH would fail. For Max Minimal Vertex Cover we give a non-trivial $\sqrt{r}$-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 super-polynomial setting: for any $\delta > 0$ it admits an $m^\delta$-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.