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 ofr for maximum independent set in $O^{\ast }(exp\left(\tilde{O}(n/r{log}^{2}r+r{log}^{2}r)\right))$ time,r for chromatic number in $O^{\ast }(exp\left(\tilde{O}(n/rlogr+r{log}^{2}r)\right))$ time, $(2-1/r)$ for minimum vertex cover in $O^{\ast }(exp(n/r^{\Omega \left(r\right)}))$ time, and $(k-1/r)$ for minimum k-hypergraph vertex cover in $O^{\ast }(exp(n/{\left(kr\right)}^{\Omega \left(kr\right)}))$ time. (Throughout, $\tilde{O}$ and $O^{\ast }$ omit $\text{polyloglog}\left(r\right)$ and factors polynomial in the input size, respectively.) The best known time bounds for all problems were $O^{\ast }\left(2^{n/r}\right)$ (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 $exp(n^{1-o\left(1\right)}/r^{1+o\left(1\right)})$ 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 $O^{\ast }\left(2^{n/r}\right)$ 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).