The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS
Andreas Emil Feldmann, Daniel Marx
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引用次数: 0

Abstract

Given a directed graph G and a list ( s 1 , t 1 ), …, ( s d , t d ) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s i → t i path for every 1 ≤ i ≤ d . The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t 1 , …, t d ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t i to every other t j ) is known to be W[1]-hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if \({\mathcal {H}} \) is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list ( s 1 , t 1 ), …, ( s d , t d ) of demands form a directed graph that is a member of \({\mathcal {H}} \) . Our main result is a complete characterization of the classes \({\mathcal {H}} \) resulting in fixed-parameter tractable special cases: we show that if every pattern in \({\mathcal {H}} \) has the combinatorial property of being “transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges,” then the problem is FPT, and it is W[1]-hard for every recursively enumerable \({\mathcal {H}} \) not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], q -Root Steiner Tree is FPT for constant q [Suchý, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.
定参数有向Steiner网络问题的复杂性格局
给定一个有向图G和一个末端对的列表(s1, t1),…,(s1, t1),有向斯坦纳网络问题要求求G的一个最小代价子图,该子图在每1≤i≤d时包含一条有向路径s1→t1。特殊情况下的有向斯坦纳树(当我们要求从根r到终端t1,…,t d的路径时)已知是由终端数量参数化的固定参数可处理的,而特殊情况下的强连通斯坦纳子图(当我们要求从每t i到每其他t j的路径时)已知是W[1]-由终端数量硬参数化的。我们系统地探索有向斯坦纳问题的复杂性景观,以充分了解哪些其他特殊情况是FPT或W[1]-困难的。形式上,如果\({\mathcal {H}} \)是一类有向图,那么我们看有向斯坦纳网络的特殊情况,其中需求的列表(s 1, t 1),…,(s d, t d)构成一个有向图,该有向图是\({\mathcal {H}} \)的成员。我们的主要结果是对类\({\mathcal {H}} \)的完整描述,导致固定参数可处理的特殊情况:我们表明,如果\({\mathcal {H}} \)中的每个模式都具有“传递等效于具有有限数量的额外边的有限长度的毛虫”的组合性质,那么问题是FPT,并且对于每个递归可枚举的\({\mathcal {H}} \)都很难不具有此性质。这一完全二分类统一并推广了已知的结果,表明有向斯坦纳树是FPT [Dreyfus and Wagner, Networks 1971], q -根斯坦纳树是常数q的FPT [Suchý, WG 2016],强连通斯坦纳子图是W[1]-hard [Guo等,SIAM J. Discrete Math. 2011],有向斯坦纳网络在常数终端数的多项式时间内可解[Feldman and Ruhl, SIAM J. Comput. 2006],而且还揭示了以前不为人知的一大片可处理病例。
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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