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URN: urn:nbn:de:0030-drops-124327
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New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures

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Abstract

In this paper, we consider the question of computing sparse subgraphs for any input directed graph G = (V,E) on n vertices and m edges, that preserves reachability and/or strong connectivity structures. - We show O(n+min{|P|√n, n√|P|}) bound on a subgraph that is an 1-fault-tolerant reachability preserver for a given vertex-pair set P ⊆ V× V, i.e., it preserves reachability between any pair of vertices in P under single edge (or vertex) failure. Our result is a significant improvement over the previous best O(n |P|) bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. - In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set P ⊆ V×V of size Ω(n^ε), for even any arbitrarily small ε, requires at least Ω(n^(1+ε/8)) edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. - We also present the first sub-quadratic bound of at most Õ(k 2^k n^(2-1/k)) size, for strong-connectivity preservers of directed graphs under k failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general k. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95].

BibTeX - Entry

@InProceedings{chakraborty_et_al:LIPIcs:2020:12432,
  author =	{Diptarka Chakraborty and Keerti Choudhary},
  title =	{{New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12432},
  URN =		{urn:nbn:de:0030-drops-124327},
  doi =		{10.4230/LIPIcs.ICALP.2020.25},
  annote =	{Keywords: Preservers, Strong-connectivity, Reachability, Fault-tolerant, Graph sparsification}
}

Keywords: Preservers, Strong-connectivity, Reachability, Fault-tolerant, Graph sparsification
Seminar: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue date: 2020
Date of publication: 29.06.2020


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