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 ofr for maximum independent set in O^*(exp ({tilde{O}}(n/r log ^2 r+rlog ^2r))) time,r for chromatic number in O^*(exp (tilde{O}(n/r log r+rlog ^2r))) time,(2-1/r) for minimum vertex cover in O^*(exp (n/r^{varOmega (r)})) time, and(k-1/r) for minimum k-hypergraph vertex cover in O^*(exp (n/ (kr)^{varOmega (kr)})) time.(Throughout, {tilde{O}} and O^* omit hbox {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 (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^*(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).
Highlights
The Independent Set, Vertex Cover, and Coloring problems are central problems in combinatorial optimization and have been extensively studied
In the setting of the more sophisticated Baker-style approximation schemes for planar graphs, Marx [34] showed that no (1 + ε)-approximating algorithm for planar Independent Set can run in time O∗(exp((1/ε)1−δ)) assuming Exponential Time Hypothesis (ETH), which implies that the algorithm of Czumaj cannot be improved to run in time O∗(exp(tw/r 1+ε))
As a final indication that sparsification is a very powerful tool to obtain fast exponential time approximation algorithms, we show that a combination of a result of Halperin [22] and the sparsification Lemma [25] gives the following result for the Vertex Cover problem in hypergraphs with edges of size at most k (a.k.a. the Set Cover problem with frequency at most k)
Summary
The Independent Set, Vertex Cover, and Coloring problems are central problems in combinatorial optimization and have been extensively studied. In the setting of the more sophisticated Baker-style approximation schemes for planar graphs, Marx [34] showed that no (1 + ε)-approximating algorithm for planar Independent Set can run in time O∗(exp((1/ε)1−δ)) assuming ETH, which implies that the algorithm of Czumaj cannot be improved to run in time O∗(exp(tw/r 1+ε)). These lower bounds, despite being interesting, do not say anything about the lower order terms and by no means answer the question whether the known approximation trade-offs can be improved significantly, and in many settings we are far from understanding the full power of exponential time approximation. We wish to design approximation algorithms that are as fast as possible
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