On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, Krzysztof Sornat

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial (32+ε)-approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

Original languageEnglish
Pages (from-to)985-1008
Number of pages24
JournalTheory of Computing Systems
Issue number6
StatePublished - 1 Aug 2021
Externally publishedYes


  • Approximation algorithms
  • Cactus augmentation
  • Connectivity augmentation
  • Cycle augmentation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics


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