NDLI: Steiner Forest Orientation Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Nutov, Zeev Kortsarz, Guy Cygan, Marek
Copyright Year	2013
Abstract	We consider connectivity problems with orientation constraints. Given a directed graph $D$ and a collection of ordered node pairs $P$ let $P[D]=\{(u,v) \in P: D \mbox{ contains a } uv\mbox{-path}\}$. In the \sf Steiner Forest Orientation problem we are given an undirected graph $G=(V,E)$ with edge-costs and a set $P \subseteq V \times V$ of ordered node pairs. The goal is to find a minimum-cost subgraph $H$ of $G$ and an orientation $D$ of $H$ such that $P[D]=P$. We give a $4$-approximation algorithm for this problem. In the \sf Maximum Pairs Orientation problem we are given a graph $G$ and a multicollection of ordered node pairs $P$ on $V$. The goal is to find an orientation $D$ of $G$ such that $\|P[D]\|$ is maximum. Generalizing the result of Arkin and Hassin [Discrete Appl. Math., 116 (2002), pp. 271--278] for $\|P\|=2$, we will show that for a mixed graph $G$ (that may have both directed and undirected edges), one can decide in $n^{O(\|P\|)}$ time whether $G$ has an orientation $D$ with $P[D]=P$. (For undirected graphs this problem admits a polynomial time algorithm for any $P$, but it is NP-complete on mixed graphs.) For undirected graphs, we will show that one can decide whether $G$ admits an orientation $D$ with $\|P[D]\| \geq k$ in $O(n+m)+2^{O(k\cdot \log \log k)}$ time; hence this decision problem is fixed-parameter tractable, which answers an open question from Dorn et al. [Algorithms Molecular Biol., 6 (2011)]. We also show that \sf Maximum Pairs Orientation admits ratio $O(\log \|P\|/\log\log \|P\|)$, which is better than the ratio $O(\log n/\log\log n)$ of Gamzu, Segev, and Sharan [Proceedings of WABI 2010, pp. 215--225] when $\|P\|
Starting Page	1503
Ending Page	1513
Page Count	11
File Format	PDF
ISSN	08954801
DOI	10.1137/120883931
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	27
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-09-12
Access Restriction	Subscribed
Subject Keyword	Nonnumerical algorithms Analysis of algorithms and problem complexity graph orientation parameterized complexity Graph theory approximation algorithms Graph algorithms Analysis of algorithms graph algorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics

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