On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

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 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 $\left (\frac {3}{2}+\varepsilon \right )$ 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.

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

“…In order to make progress on the latter problem, we study the case where the input 2-connected graph is a cycle, which we denote as the cycle vertex-connectivity augmentation problem (cycle VCA). Even for this simpler case, the best-known result is the aforementioned 2-approximation due to Auletta et al [2], while the case of augmenting the edge-connectivity of a cycle by one admits much better approximation guarantees [5], even via simple iterative algorithms [14]. Similarly to the case of edge-connectivity, it is possible to prove that cycle VCA is APX-hard (see Section 5), so our focus will be on the design of approximation algorithms.…”

“…Our algorithm is similar in spirit to the one presented by Gálvez et al [14] for the case of augmenting the edge-connectivity of a cycle; however, much more sophisticated tools are required for its analysis. In particular, the results from Gálvez et al heavily rely on the notion of contracting links (i.e.…”

“…The second case is known as the Cactus Augmentation problem, which is also APX-hard even if the cactus is a cycle [14]. For a long time the best-known approximation guarantee was 2 due to Khuller and Vishkin [18].…”

“…In this section we prove that cycle VCA is APX-hard. The following theorem summarizes a hardness result from Gálvez et al [14] in the context of edge-connectivity augmentation, stated in a way that will be useful for our purposes. Theorem 29.…”

“…The main idea of our algorithm in Section 3 is to build a collection L of circle components that, roughly speaking, use few links to cover many vertices and are pairwise non-crossing; then we add a minimal completion to obtain the desired feasible solution. A similar approach has been used by Gálvez et al for the case of edge-connectivity [14], where the authors prove that any set of links can be completed by iteratively picking and contracting links so as to reduce the whole graph into a single super-node. As mentioned before, this procedure unfortunately does not work for the case of vertex-connectivity as now we require instead to merge the different circle components into a single circle component, which is in general a strictly stronger requirement.…”