Abstract
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 cardinality 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, nothing better is known for CacAP (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 \(\mathrm {APX}\)-hard.
In this paper we present a combinatorial \(\left( \frac{3}{2}+\varepsilon \right) \)-approximation for CycAP, for any constant \(\varepsilon >0\). We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.
Partially supported by the SNSF Grant 200021_159697/1, the SNSF Excellence Grant 200020B_182865/1, the National Science Centre, Poland, grant numbers 2015/17/N/ST6/03684, 2015/18/E/ST6/00456 and 2018/28/T/ST6/00366. K. Sornat was also supported by the Foundation for Polish Science (FNP) within the START programme.
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Notes
- 1.
We recall that \(G=(V,E)\) is k-connected if for every set of edges \(F\subseteq E\), \(|F|\le k-1\), the graph \(G'=(V,E\setminus F)\) is connected.
- 2.
We recall that a cactus G is a connected undirected graph in which every edge belongs to exactly one cycle. For technical reasons it is convenient to allow cycles of length 2 consisting of parallel edges.
- 3.
A path visits a cycle iff it includes an edge from the cycle.
- 4.
This lemma implies that CycAP is FPT with parameter \(h_{\max }\).
- 5.
For \(|S|\le 1\), we simply set \(\mathrm {OPT}_S=\emptyset \).
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We would like to thank the anonymous reviewers for their helpful comments.
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Gálvez, W., Grandoni, F., Ameli, A.J., Sornat, K. (2020). On the Cycle Augmentation Problem: Hardness and Approximation Algorithms. In: Bampis, E., Megow, N. (eds) Approximation and Online Algorithms. WAOA 2019. Lecture Notes in Computer Science(), vol 11926. Springer, Cham. https://doi.org/10.1007/978-3-030-39479-0_10
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