New Tools and Connections for Exponential-Time Approximation

In this paper, we develop new tools and connections for exponential time approximation . In this setting, we are given a problem instance and an integer r>1 , and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of r for maximum independent set in O^*(exp (Õ(n/r log ^2 r+rlog ^2r))) time, r for chromatic number in O^*(exp (Õ(n/r log r+rlog ^2r))) time, (2-1/r) for minimum vertex cover in O^*(exp (n/r^ (r))) time, and (k-1/r) for minimum k -hypergraph vertex cover in O^*(exp (n/ (kr)^ (kr))) time. (Throughout, Õ and O^* omit polyloglog (r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O^*(2^n/r) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011 ; Cygan et al. in Exponential-time approximation of hard problems, 2008 ). For maximum independent set and chromatic number, these bounds were complemented by exp (n^1-o(1)/r^1+o(1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013 ; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014 ). Our results show that the naturally-looking O^*(2^n/r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule . Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32, 2016 ). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016 ; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016 ).