On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

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 men...

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Bibliographic Details
Published in:Theory of computing systems Vol. 65; no. 6; pp. 985 - 1008
Main Authors: Gálvez, Waldo, Grandoni, Fabrizio, Jabal Ameli, Afrouz, Sornat, Krzysztof
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2021
Springer Nature B.V
Subjects:
Algorithms
Approximation
Augmentation
Combinatorial analysis
Computer Science
Graph theory
Hardness
Mathematical analysis
Special Issue on Approximation and Online Algorithms
Surface hardness
Theory of Computation
Approximation algorithms
Cycle augmentation
Cactus augmentation
Connectivity augmentation
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary: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 3 2 + 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.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-020-10025-6