On subexponential running times for approximating directed Steiner tree and related problems

This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1 − α) lnn, for a given parameter 0 < α < 1. What is the best possible running time for achieving such approximation ratio? This question was answered implicitly in the work of Moshkovitz [Theory of Computing, 2015]: Assuming both the Projection Games Conjecture (PGC) and the Exponential-Time Hypothesis (ETH), any ((1 − α) lnn)-approximation algorithm for Set-Cover must run in time at least 2 c·α , for some small constant 0 < c < 1. We study the questions along this line. Our first contribution is in strengthening the above result. We show that under ETH and PGC the running time requires for any ((1−α) lnn)-approximation algorithm for Set-Cover is essentially 2 α . This (almost) settles the question since our lower bound matches the best known running time of 2 ) for approximating Set-Cover to within a factor (1 − α) lnn given by Cygan et al. [IPL, 2009]. Our result is tight up to the constant multiplying the n terms in the exponent. The lower bound of Set-Cover applies to all of its generalization, e.g., Group-Steiner-Tree, Directed-Steiner-Tree, Covering-Steiner-Tree and Connected-Polymatroid. We show that, surprisingly, in almost exponential running time, these problems reduce to Set-Cover. Specifically, we complement our lower bound by presenting an (1−α) lnn approximation algorithm for all aforementioned problems that runs in time 2 ·logn · poly(m). We further study the approximation ratio in the regime of log2−δ n for Group-Steiner-Tree and Covering-Steiner-Tree. Chekuri and Pal [FOCS, 2005] showed that Group-Steiner-Tree admits (log2−α n)-approximation in time exp(2 α+o(1) ), for any parameter 0 < α < 1. We show the running time lower bound of Group-Steiner-Tree: any (log2−α n)-approximation algorithm for GroupSteiner-Tree must run in time at least exp((1 + o(1))logα− n), for any constant > 0, unless the ETH is false. Our result follows by analyzing the hardness construction of Group-Steiner-Tree due to the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for Covering-Steiner-Tree. ∗Department of Math and information, University of Warsaw, Warsaw, Porland. Email: [email protected] †Computer Science Department, Rutgers University – Camden, Camden NJ, USA. Email: [email protected] ‡Max Planck Institute for Informatics, Saarbrücken, Germany & Institute for Theoretical Computer Science, Shanghai University of Finance and Economics, Shanghai, China. Email: [email protected]