如何隐藏小团体?

IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Uriel Feige, Vadim Grinberg
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引用次数: 0

摘要

在众所周知的 "植入小块 "问题中,一个大小为 k 的小块(或独立集)被随机植入一个 Erdos-Renyi 随机 G(n, p) 图中,目标是设计一种算法,在生成的图中找到最大的小块(或独立集)。我们在这个问题上引入了一个变种,即不是随机植入一个小块,而是由对手植入一个小块,试图让我们很难在结果图中找到最大小块。我们证明,对于问题参数的标准设置,即在随机的 \(G(n, \frac{1}{2})\) 图中植入一个大小为 \(k = \sqrt{n}\) 的簇,已知的多项式时间算法可以(以一种非微妙的方式)扩展到在对抗设置中也能工作。与此相反,我们证明,对于参数的其他自然设置,例如在一个具有 \(p = n^{-\frac{1}{2}\) 的 G(n, p) 图中种植大小为 \(k=\frac{n}{2}\) 的独立集,除非 NP 有随机多项式时间算法,否则不存在找到大小为 k 的独立集的多项式时间算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
How to Hide a Clique?

In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(np) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size \(k = \sqrt{n}\) planted in a random \(G(n, \frac{1}{2})\) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size \(k=\frac{n}{2}\) in a G(np) graph with \(p = n^{-\frac{1}{2}}\), there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms.

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来源期刊
Theory of Computing Systems
Theory of Computing Systems 工程技术-计算机:理论方法
CiteScore
1.90
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: TOCS is devoted to publishing original research from all areas of theoretical computer science, ranging from foundational areas such as computational complexity, to fundamental areas such as algorithms and data structures, to focused areas such as parallel and distributed algorithms and architectures.
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