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).

中文翻译：

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指数时间近似的新工具和连接

摘要在本文中，我们开发了用于指数时间近似的新工具和连接。在这个设置中，我们给出了一个问题实例和一个整数 $$r>1$$r>1，目标是设计一个具有最快运行时间的近似算法。我们给出的随机算法为 $$O^*(\exp ({\tilde{O}}(n/r \log ^2 r+r\log ^2r)) 中的最大独立集建立近似比率 1.r )$$O∗(exp(O~(n/rlog2r+rlog2r))) 时间，2.r 表示 $$O^*(\exp (\tilde{O}(n/r \log r+ r\log ^2r)))$$O∗(exp(O~(n/rlogr+rlog2r))) 时间，3.$$(2-1/r)$$(2-1/r) 为最小值$$O^*(\exp (n/r^{\varOmega (r)}))$$O∗(exp(n/rΩ(r))) 时间的顶点覆盖，以及4.$$(k-1 /r)$$(k-1/r) 用于 $$O^*(\exp (n/ (kr)^{\varOmega (kr)}))$$O∗(exp (n/(kr)Ω(kr))) 时间。（始终，$${\tilde{O}}$$O~ 和 $$O^*$$O∗ 分别省略了输入大小中的 $$\hbox {polyloglog} (r)$$$polyloglog(r) 和因子多项式。 ) 所有问题最著名的时间界限是 $$O^*(2^{n/r})$$O*(2n/r)（Bourgeois 等人在 Discret Appl Math 159(17):1954–1970 中） , 2011 年；Cygan 等人在困难问题的指数时间近似中，2008 年）。对于最大独立集和色数，这些边界由 $$\exp (n^{1-o(1)}/r^{1+o(1)})$$exp(n1-o(1) /r1+o(1)) 下界（在指数时间假设 (ETH) 下）（Chalermsook 等人在计算机科学基础，FOCS，第 370-379 页，2013 年；Laekhanukit 在次指数中组合问题的不可近似性-时间。博士论文，2014）。我们的结果表明，对于所有这些问题，看起来自然的 $$O^*(2^{n/r})$$O∗(2n/r) 边界并不严格。这些结果的关键是将问题简化为有界度变体的稀疏化过程，允许对有界度图使用近似算法。为了获得前两个结果，我们引入了一个新的随机分支规则。最后，我们展示了 PCP 参数和指数时间近似算法之间的联系。这种联系与我们的独立集算法一起驳斥了过度减小 Chan 的 PCP 大小的可能性（Chan in J. ACM 63(3):27:1–27:32, 2016）。这也意味着对我们结果的（显着）改进将驳斥 gap-ETH 猜想（Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016；Manurangsi 和 Raghavendra 在 Abirthday重复定理和近似密集 CSP 的复杂性中， 2016）。