## Abstract

Let G = (V, E) be an n-vertices m-edges directed graph. Let s ∈ V be any designated source vertex. We address the problem of single source reachability (SSR) from s in presence of failures of vertices/edges. We show that for every k ≥ 1, there is a subgraph H of G with at most 2^{k} n 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 u ∈ V is reachable from s in G \ F if and only if u is reachable from s in H \ F. We call H a k-Fault Tolerant Reachability Subgraph (k-FTRS). We prove also a matching lower bound of Ω(2^{k} n) for such subgraphs. Our results extend to vertex failures without any extra overhead. The general 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 graphs, there is a lower bound of Ω(m) even for a single edge failure [9]. In the case of unweighted graphs there is a lower bound of Ω(n^{3/2}) edges, again, even for a single edge failure [16]. There is also a matching upper bound but nothing is known for two or more failures in the directed graphs. 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, (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 [12] in their pioneering work on flows and cuts. 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.

Original language | English |
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Title of host publication | STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Yishay Mansour, Daniel Wichs |

Publisher | Association for Computing Machinery |

Pages | 509-518 |

Number of pages | 10 |

ISBN (Electronic) | 9781450341325 |

DOIs | |

State | Published - 19 Jun 2016 |

Event | 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 - Cambridge, United States Duration: 19 Jun 2016 → 21 Jun 2016 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 19-21-June-2016 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 |
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Country/Territory | United States |

City | Cambridge |

Period | 19/06/16 → 21/06/16 |

### Bibliographical note

Funding Information:This research was partially supported by Israel Science Foundation (ISF) and University Grants Commission (UGC) of India. Additionally, the research of the first author was partially supported by Indo-German Max-Planck Center for Computer Science (IMPECS), and the research of the second author was partially supported by Google India under the Google India PhD Fellowship Award.

Publisher Copyright:

© 2016 ACM.

## Keywords

- Farthest min-cut
- Fault Tolerant
- Single-source reachability