New Tools and Connections for Exponential-Time Approximation

AbstractIn this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer $$r>1$$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 of1.r for maximum independent set in $$O^*(\exp ({\tilde{O}}(n/r \log ^2 r+r\log ^2r)))$$O∗(exp(O~(n/rlog2r+rlog2r))) time,2.r for chromatic number in $$O^*(\exp (\tilde{O}(n/r \log r+r\log ^2r)))$$O∗(exp(O~(n/rlogr+rlog2r))) time,3.$$(2-1/r)$$(2-1/r) for minimum vertex cover in $$O^*(\exp (n/r^{\varOmega (r)}))$$O∗(exp(n/rΩ(r))) time, and4.$$(k-1/r)$$(k-1/r) for minimum k-hypergraph vertex cover in $$O^*(\exp (n/ (kr)^{\varOmega (kr)}))$$O∗(exp(n/(kr)Ω(kr))) time. (Throughout, $${\tilde{O}}$$O~ and $$O^*$$O∗ omit $$\hbox {polyloglog} (r)$$polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were $$O^*(2^{n/r})$$O∗(2n/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 $$\exp (n^{1-o(1)}/r^{1+o(1)})$$exp(n1-o(1)/r1+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 $$O^*(2^{n/r})$$O∗(2n/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).