arXiv 1707.010372017-07-04cs.DScs.DM

包装周期比Erdens-Pósa更快

Packing Cycles Faster Than Erdős-Pósa

Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh, Meirav Zehavi

摘要：

循环包装问题询问给定的无向图$G=(V,E)$是否包含$k$顶点不相交的循环。自1965年经典的Erdens-Pósa定理发表以来，这个问题在图论和算法设计领域受到了极大的科学关注。特别是，这个问题是在参数化复杂性的框架中研究的第一个问题之一。循环填充的非均匀固定参数易处理性来自罗伯逊-西摩定理，这是Fellows和Langston在20世纪80年代已经观察到的事实。在1994年，Bodlaender证明了可以使用指数空间在时间$2^{\mathcal{O}(k^2)}\cdot |V|$中求解循环填充。如果存在解，Bodlaender的算法也会输出解（同时）。它后来成为常识，循环包装承认$2^{\mathcal{O}(k\log^2k)}\cdot |V|$ -时间（确定性）算法使用指数空间，这是一个后果的Erdens-Pósa定理。目前，该算法的设计在有关参数化复杂性的教科书中作为练习给出。然而，还没有找到在时间上运行$2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$的算法，击败了约束$2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$。鉴于此，询问$2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$约束是否本质上是最优的似乎是很自然的。在本文中，我们回答这个问题的负面发展的$2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|$时间（确定性）算法循环包装。如果存在解，我们的算法也输出一个解（同时）。此外，除了击败界$2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$，我们的算法运行在时间上的线性$|V|$，其空间复杂度是多项式的输入大小。

英文摘要：

The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdős-Pósa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson-Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time $2^{\mathcal{O}(k^2)}\cdot |V|$ using exponential space. In case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a $2^{\mathcal{O}(k\log^2k)}\cdot |V|$-time (deterministic) algorithm using exponential space, which is a consequence of the Erdős-Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time $2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, has been found. In light of this, it seems natural to ask whether the $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a $2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|$-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, our algorithm runs in time linear in $|V|$, and its space complexity is polynomial in the input size.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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