New tools and connections for exponential-time approximation

Nikhil Bansal, Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai, Jesper Nederlof (Corresponding author)

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Abstract

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~ (n/ rlog 2r+ rlog 2r))) time,2.r for chromatic number in O(exp (O~ (n/ rlog r+ rlog 2r))) time,3.(2 - 1 / r) for minimum vertex cover in O(exp (n/ rΩ ( r ))) time, and4.(k- 1 / r) for minimum k-hypergraph vertex cover in O(exp (n/ (kr) Ω ( k r ))) time. (Throughout, O~ 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 (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) 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).

Original languageEnglish
Pages (from-to)3993-4009
Number of pages17
JournalAlgorithmica
Volume81
Issue number10
DOIs
Publication statusPublished - 1 Oct 2019

Bibliographical note

Part of the following topical collections:•Special Issue: Parameterized and Exact Computation

Keywords

  • Approximation algorithms
  • Exponential time algorithms
  • PCP’s
  • PCP's

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