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

在k连通性增强问题中，我们得到了k边缘连通图和一组称为link的附加边。我们的目标是找到一组最小尺寸的链​​接，将其添加到图形中使其成为（k +1）边连接。从所提到的问题到k = 1（又名树增强问题或TAP）或k = 2（又名仙人掌增强问题或CacAP）的情况，可以近似地保留减少量。虽然TAP有几种优于2的近似算法，但对于CacAP直到最近才突破了这一障碍（因此k-一般情况下的连接性增强）。作为朝着CacAP更好的近似算法迈出的第一步，我们考虑了特殊情况，其中输入仙人掌由单个循环组成，即循环增强问题（CycAP）。这种看似简单的特殊情况保留了普通情况下的部分硬度。特别是，我们能够证明它是APX认证的。在本文中，对于任何常数ε > 0，我们给出了CycAP的组合\（\ left（\ frac {3} {2} + \ varepsilon \ right）\） -逼近。差距：这可能有助于解决问题的一般情况。