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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2020.25
URN: urn:nbn:de:0030-drops-124327
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12432/
Chakraborty, Diptarka ;
Choudhary, Keerti
New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures
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}
}
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Keywords: |
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Preservers, Strong-connectivity, Reachability, Fault-tolerant, Graph sparsification |
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Collection: |
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47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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29.06.2020 |
