{"title":"FPT Algorithms to Enumerate and Count Acyclic and Totally Cyclic Orientations","authors":"Farley Soares Oliveira, Hidefumi Hiraishi, Hiroshi Imai","doi":"10.1016/j.entcs.2019.08.057","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we deal with counting and enumerating problems for two types of graph orientations: acyclic and totally cyclic orientations. Counting is known to be #P-hard for both of them. To circumvent this issue, we propose Fixed Parameter Tractable (FPT) algorithms. For the enumeration task, we construct a Binary Decision Diagram (BDD) to represent all orientations of the two kinds, instead of explicitly enumerating them. We prove that the running time of this construction is bounded by <em>O*</em>(2<sup>pw<sup>2</sup>/4+<em>o</em>(pw<sup>2</sup>)</sup>) with respect to the pathwidth pw. We then develop faster FPT algorithms to count acyclic and totally acyclic orientations, running in <em>O*</em>(2<sup>bw<sup>2</sup>/2+o(bw<sup>2</sup>)</sup>) time, where bw denotes the branch-width of the given graph. These counting algorithms are obtained by applying the observations in our enumerating algorithm to branch decomposition.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":"346 ","pages":"Pages 655-666"},"PeriodicalIF":0.0000,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.057","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Notes in Theoretical Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1571066119301082","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Computer Science","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we deal with counting and enumerating problems for two types of graph orientations: acyclic and totally cyclic orientations. Counting is known to be #P-hard for both of them. To circumvent this issue, we propose Fixed Parameter Tractable (FPT) algorithms. For the enumeration task, we construct a Binary Decision Diagram (BDD) to represent all orientations of the two kinds, instead of explicitly enumerating them. We prove that the running time of this construction is bounded by O*(2pw2/4+o(pw2)) with respect to the pathwidth pw. We then develop faster FPT algorithms to count acyclic and totally acyclic orientations, running in O*(2bw2/2+o(bw2)) time, where bw denotes the branch-width of the given graph. These counting algorithms are obtained by applying the observations in our enumerating algorithm to branch decomposition.
本文讨论了两类图的取向:无环取向和全循环取向的计数和枚举问题。众所周知，计数对他们两人来说都很困难。为了避免这个问题，我们提出了固定参数可处理(FPT)算法。对于枚举任务，我们构造了一个二进制决策图(BDD)来表示两种类型的所有方向，而不是显式枚举它们。我们证明了这种结构的运行时间以O*(2pw2/4+ O (pw2))关于路径宽度pw为界。然后，我们开发了更快的FPT算法来计算无环和完全无环方向，运行时间为O*(2bw2/2+ O (bw2))，其中bw表示给定图的分支宽度。这些计数算法是通过将枚举算法中的观察结果应用于分支分解而得到的。
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