On the Complexity of Finding a Sparse Connected Spanning Subgraph in a Non-Uniform Failure Model

Matthias Bentert, Jannik Schestag, Frank Sommer

Research output: Chapter in Book/Conference proceedings/Edited volumeConference contributionScientificpeer-review

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Abstract

We study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either safe or unsafe and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph G = (V,E) in which the edge set E is partitioned into a set S of safe edges and a set U of unsafe edges and the task is to find a set T of at most k edges such that T -{u} is connected and spans V for any unsafe edge u ∈ T. Unweighted Flexible Graph Connectivity generalizes both Spanning Tree and Hamiltonian Cycle. We study Unweighted Flexible Graph Connectivity in terms of fixed-parameter tractability (FPT). We show an almost complete dichotomy on which parameters lead to fixed-parameter tractability and which lead to hardness. To this end, we obtain FPT-time algorithms with respect to the vertex deletion distance to cluster graphs and with respect to the treewidth. By exploiting the close relationship to Hamiltonian Cycle, we show that FPT-time algorithms for many smaller parameters are unlikely under standard parameterized complexity assumptions. Regarding problem-specific parameters, we observe that Unweighted Flexible Graph Connectivity admits an FPT-time algorithm when parameterized by the number of unsafe edges. Furthermore, we investigate a below-upper-bound parameter for the number of edges of a solution. We show that this parameter also leads to an FPT-time algorithm.

Original languageEnglish
Title of host publication18th International Symposium on Parameterized and Exact Computation, IPEC 2023
EditorsNeeldhara Misra, Magnus Wahlstrom
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages12
ISBN (Electronic)9783959773058
DOIs
Publication statusPublished - 2023
Event18th International Symposium on Parameterized and Exact Computation, IPEC 2023 - Amsterdam, Netherlands
Duration: 6 Sept 20238 Sept 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume285
ISSN (Print)1868-8969

Conference

Conference18th International Symposium on Parameterized and Exact Computation, IPEC 2023
Country/TerritoryNetherlands
CityAmsterdam
Period6/09/238/09/23

Keywords

  • below-guarantee parameterization
  • Flexible graph connectivity
  • NP-hard problem
  • parameterized complexity
  • treewidth

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