New Tools And Connections For Exponential-time Approximation
September 11, 2019 - Bansal, Nikhil; Chalermsook, Parinya; Laekhanukit, Bundit; Nanongkai, Danupon; Nederlof, Jesper
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EmailJournal or Book Title: ALGORITHMICA
Volume/Issue: 81
Year Published: 2019
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 of1. r for maximum independent set in O*(exp((O) over tilde (n/r log(2) r + r log(2) r))) time,2. r for chromatic number in O*(exp((O) over tilde (n/r log r + r log(2) r))) time,3. (2 - 1/r) for minimum vertex cover in O*(exp(n/r(Omega(r)))) time, and4. (k - 1/r) for minimum k-hypergraph vertex cover in O*(exp(n/(kr)(Omega(kr)))) time.(Throughout, (O) over tilde and O* omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were 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))) 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*(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).
DOI: 10.1007/s00453-018-0512-8
Type of Publication: Article