## Abstract

One of the most important and well-studied settings for network design is edge-connectivity requirements. This encompasses uniform demands such as the Minimum k-Edge-Connected Spanning Subgraph problem (k-ECSS), as well as nonuniform demands such as the Survivable Network Design problem. A weakness of these formulations, though, is that we are not able to ask for fault-tolerance larger than the connectivity. Taking inspiration from recent definitions and progress in graph spanners, we introduce and study new variants of these problems under a notion of relative fault-tolerance. Informally, we require not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph post-faults. That is, the subgraph we build must “behave” identically to the underlying graph with respect to connectivity after bounded faults. We define and introduce these problems, and provide the first approximation algorithms: a (1 + 4/k)-approximation for the unweighted relative version of k-ECSS, a 2-approximation for the weighted relative version of k-ECSS, and a 27/4-approximation for the special case of Relative Survivable Network Design with only a single demand with a connectivity requirement of 3. To obtain these results, we introduce a number of technical ideas that may of independent interest. First, we give a generalization of Jain's iterative rounding analysis that works even when the cut-requirement function is not weakly supermodular, but instead satisfies a weaker definition we introduce and term local weak supermodularity. Second, we prove a structure theorem and design an approximation algorithm utilizing a new decomposition based on important separators, which are structures commonly used in fixed-parameter algorithms that have not commonly been used in approximation algorithms.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022 |

Editors | Amit Chakrabarti, Chaitanya Swamy |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 41:1-41:19 |

Number of pages | 19 |

ISBN (Electronic) | 9783959772495 |

DOIs | |

State | Published - 1 Sep 2022 |

Externally published | Yes |

Event | 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 - Virtual, Urbana-Champaign, United States Duration: 19 Sep 2022 → 21 Sep 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 245 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 |
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Country/Territory | United States |

City | Virtual, Urbana-Champaign |

Period | 19/09/22 → 21/09/22 |

### Bibliographical note

Publisher Copyright:© Michael Dinitz, Ama Koranteng, and Guy Kortsarz.

## Keywords

- Fault Tolerance
- Network Design