NDLI: Fault-Tolerant Subgraph for Single-Source Reachability: General and Optimal

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Fault-Tolerant Subgraph for Single-Source Reachability: General and Optimal

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Baswana, Surender Choudhary, Keerti Roditty, Liam
Copyright Year	2018
Abstract	Let $G$ be a directed graph with $n$ vertices, $m$ edges, and a designated source vertex $s$. We address the problem of single-source reachability (SSR) from $s$ in the presence of failures of vertices/edges. We show that for every $k\geq1$, there is a subgraph $H$ of $G$ with at most $2^kn$ edges that preserves the reachability from $s$ even after the failure of any $k$ edges. Formally, given a set $F$ of $k$ edges, a vertex $v\in V(G)$ is reachable from $s$ in $G\setminus F$ if and only if $v$ is reachable from $s$ in $H\setminus F$. We call $H$ a $k$-fault tolerant reachability subgraph ($\textsc{$k$-FTRS}$). We also prove a matching lower bound of $\Omega(2^kn)$ edges for such subgraphs that holds for all $n,k$ with $2^k\leq n$. Our results extend to vertex failures without any extra overhead. The construction of ${$k$-FTRS}$ is interesting from several different perspectives. From the Graph theory perspective it reveals a separation between SSR and single-source shortest paths (SSSP) in directed graphs. More specifically, in the case of SSSP in weighted directed and undirected graphs, Demetrescu et al. showed that there is a lower bound of $\Omega(m)$ edges even for a single edge failure [SIAM J. Comput., 37 (2008), pp. 1299--1318]. In the case of unweighted graphs Parter and Peleg gave a lower bound of $\Omega(n^{3/2})$ edges, again, even for a single edge failure [Proc. Algorithms---21st Annual European Symposium, 2013, pp. 779--790]. From the Algorithms perspective it implies fault-tolerant algorithms for other interesting problems, namely, (i) verifying if the strong connectivity of a graph is preserved after $k$ edge or vertex failures, and (ii) computing a dominator tree of a graph after $k$-failures. From the perspective of techniques it makes an interesting usage of the concept of farthest min-cut which was already introduced by Ford and Fulkerson in their pioneering work on flows and cuts [Flows in Networks, Princeton University Press, 1962; reprinted 2011]. We show that there is a close relationship between the farthest min-cut and the ${$k$-FTRS}$. We believe that our new technique is of independent interest.
Sponsorship	University Grants Commission. Google India. Israel Science Foundation
Starting Page	80
Ending Page	95
Page Count	16
File Format	PDF
ISSN	00975397
DOI	10.1137/16M1087643
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	1
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-01-11
Access Restriction	Subscribed
Subject Keyword	Nonnumerical algorithms subgraph Directed graphs (digraphs), tournaments reachability Graph algorithms Analysis of algorithms fault-tolerant min-cut
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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