通过限定k步法更快地找到偶数周期

Søren Dahlgaard, M. B. T. Knudsen, Morten Stöckel
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引用次数: 20

摘要

图中求环是算法图论中的一个基本问题。本文考虑在一个有n个节点和m条边且k≥2的无向图G中寻找和报告一个长度为2k的循环的问题。Bondy和Simonovits的一个经典结果[J]。组合理论,1974]表明,如果m≥100k n1+1/k,则G包含一个2k周期,进一步表明只需考虑m = O(n1+1/k)的图。以前最著名的算法是由Yuster和Zwick提出的O(n2)算法[J]。离散数学1997]以及由Alon等人[Algorithmica 1997]编写的O(m2-(1+ (k /2²)-1)/(k+1))算法。我们提出了一个算法,该算法使用O(m2k/(k+1))时间并找到一个2k周期(如果存在)。当m = Θ(n1+1/k)时,这个边界是O(n2)当发现4个循环时,我们的新边界与Alon等人一致,而对于每一个k>2,我们的新边界在m上产生一个多项式改进。Yuster和Zwick指出,“推测O(n2)是关于n的最佳可能边界是合理的”。我们展示了“条件最优性”:如果这个假设成立,那么我们的O(m2k/(k+1))算法也是严密的。此外,一个民间约简表明,没有组合算法可以确定一个图是否在时间O(m3/2-ε)上包含一个6周期,对于任何ε>0,除非布尔矩阵乘法可以在时间O(n3-ε ')上对某些ε ' >0进行组合求解,这被广泛认为是错误的。结合我们的主要结果,这给出了寻找6个周期组合的严格界限,也分离了寻找4和6个周期的复杂性,这证明了运行时间中m的指数确实应该随着k的增加而增加。我们算法的关键成分是一个新的概念,即长度为k的行走,根据固定的顺序只访问节点。我们的主要技术贡献是一个复杂的分析,证明了这种行走的几个特性,这些特性可能是独立的兴趣。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Finding even cycles faster via capped k-walks
Finding cycles in graphs is a fundamental problem in algorithmic graph theory. In this paper, we consider the problem of finding and reporting a cycle of length 2k in an undirected graph G with n nodes and m edges for constant k≥ 2. A classic result by Bondy and Simonovits [J. Combinatorial Theory, 1974] implies that if m ≥ 100k n1+1/k, then G contains a 2k-cycle, further implying that one needs to consider only graphs with m = O(n1+1/k). Previously the best known algorithms were an O(n2) algorithm due to Yuster and Zwick [J. Discrete Math 1997] as well as a O(m2-(1+⌈ k/2 ⌉-1)/(k+1)) algorithm by Alon et. al. [Algorithmica 1997]. We present an algorithm that uses O( m2k/(k+1) ) time and finds a 2k-cycle if one exists. This bound is O(n2) exactly when m = Θ(n1+1/k). When finding 4-cycles our new bound coincides with Alon et. al., while for every k>2 our new bound yields a polynomial improvement in m. Yuster and Zwick noted that it is "plausible to conjecture that O(n2) is the best possible bound in terms of n". We show "conditional optimality": if this hypothesis holds then our O(m2k/(k+1)) algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a 6-cycle in time O(m3/2-ε) for any ε>0 unless boolean matrix multiplication can be solved combinatorially in time O(n3-ε′) for some ε′ > 0, which is widely believed to be false. Coupled with our main result, this gives tight bounds for finding 6-cycles combinatorially and also separates the complexity of finding 4- and 6-cycles giving evidence that the exponent of m in the running time should indeed increase with k. The key ingredient in our algorithm is a new notion of capped k-walks, which are walks of length k that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.
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