New Tools and Connections for Exponential-time Approximation

TL;DR

This paper introduces new exponential-time approximation algorithms for problems like maximum independent set, chromatic number, and vertex cover, surpassing previous bounds and linking PCP parameters to approximation limits.

Contribution

It presents novel algorithms with improved exponential bounds and establishes a connection between PCP parameters and exponential-time approximation complexity.

Findings

Improved exponential bounds for maximum independent set and chromatic number.

New randomized branching rule for approximation algorithms.

Refutation of overly optimistic PCP size reductions based on these results.

Abstract

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and a parameter $\alpha>1$, and the goal is to design an $\alpha$-approximation algorithm with the fastest possible running time. We show the following results: - An $r$-approximation for maximum independent set in $O^*(\exp(\tilde O(n/r \log^2 r+r\log^2r)))$ time, - An $r$-approximation for chromatic number in $O^*(\exp(\tilde{O}(n/r \log r+r\log^2r)))$ time, - A $(2-1/r)$-approximation for minimum vertex cover in $O^*(\exp(n/r^{\Omega(r)}))$ time, and - A $(k-1/r)$-approximation for minimum $k$-hypergraph vertex cover in $O^*(\exp(n/(kr)^{\Omega(kr)}))$ time. (Throughout, $\tilde O$ and $O^*$ omit $\mathrm{polyloglog}(r)$ and factors polynomial in the input size, respectively.) The best known time bounds for all problems were $O^*(2^{n/r})$ [Bourgeois et al. 2009, 2011 & Cygan et al. 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., 2013 & Laekhanukit, 2014 (Ph.D. Thesis)]. Our results show that the naturally-looking $O^*(2^{n/r})$ bounds are not tight for all these problems. The key to these algorithmic results is a sparsification procedure, allowing the use of better approximation algorithms for bounded degree graphs. For obtaining 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, 2016]. It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture [Dinur 2016 & Manurangsi and Raghavendra, 2016].