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Approximation Algorithms for Minimum-Cost $k\hbox{-}(S,T)$ Connected Digraphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Laekhanukit, B. Cheriyan, J.
Copyright Year	2013
Abstract	In the minimum-cost $k\hbox{-}(S,T)$ connected digraph (abbreviated as $k\hbox{-}(S,T)$ connectivity) problem we are given a positive integer $k$, a directed graph $G=(V,E)$ with nonnegative costs on the edges, and two subsets $S,T$ of $V$; the goal is to find a subset of edges $\widehat{E}$ of minimum cost such that the subgraph $(V,\widehat{E})$ has $k$ edge-disjoint directed paths from each vertex in $S$ to each vertex in $T$. Most of our results focus on a specialized version of the problem that we call the standard version, where every edge of positive cost has its tail in $S$ and its head in $T$. This version of the problem captures NP-hard problems such as the minimum-cost $k$-vertex connected spanning subgraph problem. We give an approximation algorithm with a guarantee of $O((\log{k})(\log{n}))$ for the standard version of the $k\hbox{-}(S,T)$ connectivity problem, where $n$ denotes the number of vertices. For $k=1$, we give a simple 2-approximation algorithm that generalizes a well-known 2-approximation algorithm for the minimum-cost strongly connected spanning subgraph problem. For $k=2$, we give a 3-approximation algorithm; this matches the best approximation guarantee known for the special case of the minimum-cost $2$-vertex connected spanning subgraph problem. Besides the standard version, we study another version that is intermediate between the standard version and the problem in its full generality. In the relaxed version of the $(S,T)$ connectivity problem, each edge of positive cost has its head in $T$ but there is no restriction on the tail. We study the relaxed version with the connectivity parameter $k$ fixed at one and observe that this version is at least as hard to approximate as the directed Steiner tree problem. We match this by giving an algorithm that achieves an approximation guarantee of $\alpha(n)+1$ for the relaxed $(S,T)$ connectivity problem, where $\alpha(n)$ denotes the best approximation guarantee available for the directed Steiner tree problem. The key to the analysis is a structural result that decomposes any feasible solution into a set of so-called junction trees that are disjoint on the vertices of $T$. Our algorithm and analysis specialize to the case when the input digraph is acyclic on $T$, meaning that there exists no dicycle that contains two distinct vertices of $T$. In this setting, we show that the relaxed $(S,T)$ connectivity problem is at least as hard to approximate as the set covering problem, and we prove that our algorithm achieves a matching approximation guarantee of $O(\log{\|S\|})$.
Starting Page	1450
Ending Page	1481
Page Count	32
File Format	PDF
ISSN	08954801
DOI	10.1137/100818728
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	2013-08-22
Access Restriction	Subscribed
Subject Keyword	directed Steiner tree Flows in graphs approximation algorithms Combinatorial optimization rooted connectivity Programming involving graphs or networks graph connectivity Connectivity Approximation algorithms Directed graphs (digraphs), tournaments $k$-vertex connected spanning subgraphs Graph algorithms network design
Content Type	Text
Resource Type	Article
Subject	Mathematics

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