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New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures

Authors Diptarka Chakraborty, Keerti Choudhary

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Author Details

Diptarka Chakraborty
  • National University of Singapore, Singapore
Keerti Choudhary
  • Tel Aviv University, Israel

Funding

  • Chakraborty, Diptarka: Supported in part by NUS ODPRT Grant, WBS No. R-252-000-A94-133.
  • Choudhary, Keerti: Supported in part by ERC Grant No. 803118.

Acknowledgements

Authors would like to thank Robert Krauthgamer and Greg Bodwin for some useful discussion.

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Diptarka Chakraborty and Keerti Choudhary. New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.25

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].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Preservers
  • Strong-connectivity
  • Reachability
  • Fault-tolerant
  • Graph sparsification

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