Parameterized algorithms for generalizations of Directed Feedback Vertex Set

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Alexander Göke , Dániel Marx , Matthias Mnich
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引用次数: 0

Abstract

The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph G and seeks a smallest vertex set S that hits all cycles in G. This is one of Karp’s 21 NP-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a 4kk!nO(1)-time algorithm, where k=|S|.

Here we show fixed-parameter tractability of two generalizations of DFVS:

  • Find a smallest vertex set S such that every strong component of GS has size at most s: we give an algorithm solving this problem in time 4k(ks+k+s)!nO(1). This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.

  • Find a smallest vertex set S such that every non-trivial strong component of GS is 1-out-regular: we give an algorithm solving this problem in time 2O(k3)nO(1).

We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.
有向反馈顶点集泛化的参数化算法
有向反馈顶点集(DFVS)问题以一个有向图G作为输入,并寻找一个最小的顶点集S,该顶点集S可以到达G中的所有循环。这是Karp的21个np完全问题之一。求解DFVS的参数化复杂性状态是一个长期存在的开放性问题,直到Chen等人(2008)通过k=|S|的4kk!nO(1)时间算法显示其固定参数可追溯性。•找到一个最小顶点集S,使得G−S的每个强分量的大小不超过S,并给出一个算法,在4k(ks+k+ S)!这推广了Xiao(2017)针对该问题的无向版本的算法。•找到一个最小的顶点集S,使得G−S的每个非平凡强分量都是1-外正则的:我们给出了一个在2O(k3)⋅nO(1)时间内解决这个问题的算法。我们还用定参数算法求解了这些问题的相应弧线版本。
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来源期刊
Discrete Optimization
Discrete Optimization 管理科学-应用数学
CiteScore
2.10
自引率
9.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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