Strong Inapproximability of the Basic k-Spanner Problem
Strong Inapproximability of the Basic k-Spanner Problem
复制标题
基本 k-Spanner 问题的强不可逼近性
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
D. Peleg
中科院分区：
文献类型：
--
作者：
Michael Elkin;D. Peleg
本文研究了稀疏的K-Spanner问题的近似性。 /⌊k⌋） - approximation比率算法[14]。对于每个k≥2[11]，对于K = 2，该下限都很紧。
本文考虑到任何常数k> 2的问题的强大（或III级[10]）通过提出强大的（或III类）的不Xibibibibibility结果来缩小差距，即，对于任何o（2logɛn），该问题表明该问题是不合适的（2logɛn）固定0 <<1，除非np⊆dimie（npolylog n）。
该硬度结果延伸到o（2logɛn）的结果 - 对于k = logµ n和0 <ɛ<1- µ的k-spanner问题的易X型，对于任何0 <µ <1。 o（2log1-µ n） - 对问题的算法所隐含的算法，对于我们所知，这是第一个III类问题的示例。从这个意义上讲，上限和下限“汇聚”。
我们的主要结果也意味着对于该问题的其他一些变体也有同样的硬度，这些变体以前尚不清楚，例如统一的K-Spanner问题，单位重量K-Spanner问题，三翼型跨学家的增强问题和“三翼型k-Spanner问题”和“所有常数k的全服务器“ k-spanner问题。
This paper studies the approximability of the sparse k-spanner problem. An O(log n)-ratio approximation algorithm is known for the problem for k = 2. For larger values of k, the problem admits only a weaker O(n1/⌊k⌋)-approximation ratio algorithm [14]. On the negative side, it is known that the k-spanner problem is weakly inapproximable, namely, it is NP-hard to approximate the problem with ratio O(log n), for every k ≥ 2 [11]. This lower bound is tight for k = 2 but leaves a considerable gap for small constants k > 2.
This paper considerably narrows the gap by presenting a strong (or Class III [10]) inapproximability result for the problem for any constant k > 2, namely, showing that the problem is inapproximable within a ratio of O(2logƐ n), for any fixed 0 < Ɛ < 1, unless NP ⊆ DTIME(npolylog n). Hence the k-spanner problem exhibits a "jump" in its inapproximability once the required stretch is increased from k = 2 to k = 2+δ.
This hardness result extends into a result of O(2logƐ n)-inapproximability for the k-spanner problem for k = logµ n and 0 < Ɛ < 1 - µ, for any 0 < µ < 1. This result is tight, in view of the O(2log1-µ n)-approximation ratio for the problem, implied by the algorithm of [14] for the case k = logµ n. To the best of our knowledge, this is the first example for a set of Class III problems for which the upper and lower bounds "converge" in this sense.
Our main result implies also the same hardness for some other variants of the problem whose strong inapproximability was not known before, such as the uniform k-spanner problem, the unit-weight k-spanner problem, the 3-spanner augmentation problem and the "all-server" k-spanner problem for any constant k.