定参数有向Steiner网络问题的复杂性格局

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS
A. Feldmann, D. Marx
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引用次数: 18

摘要

给定有向图G和终端对的列表(s1,t1),…,(sd,td),有向Steiner网络问题要求G的最小代价子图包含有向si→ 每1≤i≤d的ti路径。已知特例有向施泰纳树(当我们要求从根r到终端t1,…,td的路径时)是可通过终端数量参数化的固定参数,而特殊情况下的强连通Steiner子图(当我们要求从每一个ti到每一个其他tj的路径时)已知是由终端的数量来硬参数化的W[1]。我们系统地探索了有向Steiner问题的复杂性景观,以充分理解其他哪些特殊情况是FPT或W[1]-困难的。形式上,如果\({\mathcal{H}})是一类有向图,那么我们研究有向Steiner网络的特殊情况,其中需求的列表(s1,t1),…,(sd,td)形成了一个有向图。我们的主要结果是类\({\mathcal{H}}\)的完全刻画,导致固定参数可处理的特殊情况:我们证明了如果\({\mathcal{H}}\)中的每个模式都具有“传递等价于具有有界数量的额外边的有界长度毛虫”的组合性质,那么问题是FPT,并且对于每一个不具有此性质的递归可枚举\({\mathcal{H}}),它是W[1]-硬的。这种完全二分法统一并推广了已知结果,表明有向Steiner树是FPT[Dreyfus和Wagner,Networks 1971],q-Root Steiner树对常数q[Ssuchý,WG 2016]是FPT,强连通Steiner子图是W[1]-hard[Guo et al.,SIAM J.Discrete Math.2011],并且对于恒定数量的终端,有向Steiner网络在多项式时间内是可解的[Feldman和Ruhl,SIAM J.Comput.2006],并且还揭示了以前未知的大量可处理的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems
Given a directed graph G and a list (s1, t1), …, (sd, td) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ d. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, …, td) 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 ti to every other tj) 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 (s1, t1), …, (sd, td) 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.
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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
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发文量
10
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