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The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem (CAP) is arguably one of the most basic problems in this area: given a k(-edge)-connected graph G and a set of extra edges (links), select a minimum cardinality subset A of links such that adding A to G increases its edge connectivity to k+1. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and J\'aj\'a, SIAM J. Comput., 10 (1981), pp. 270--283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290--306] that CAP can be reduced to the case k = 1, also known as the tree augmentation problem (TAP) for odd k, and to the case k = 2, also known as the cactus augmentation problem (CacAP) for even k. Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC'20, ACM, New York, 2020, pp. 815--825], several better than 2 approximation algorithms were known for TAP, culminating with a recent 1.458 approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC'18, ACM, New York, 1918, pp. 632--645]. However, for CacAP the best known approximation was 2. In this paper we breach the 2 approximation barrier for CacAP, hence, for CAP, by presenting a polynomial-time 2 ln(4) 967 1120 + \varepsilon < 1.91 approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for TAP either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP'14, Springer, Berlin, 2014, pp. 800--811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we "open the box" and exploit the specific properties of the instances of a Steiner tree arising from CacAP. In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area. [ABSTRACT FROM AUTHOR]

Obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) construction is an essential part of routing. Recently, IC routing and related researches have been extended from Manhattan architecture (λ2-geometry) to Y-/X-architecture (λ3-/λ4-geometry) to improve the chip performance. This paper presents an O(n log n) heuristic, λ-OAT, for obstacle-avoiding Steiner minimal tree construction in the A-geometry plane (λ-OASMT). In this paper, based on obstacle-avoiding constrained Delaunay triangulation, a full connected tree is constructed and then embedded into λ-OASMT by zonal combination. To the best of our knowledge, this is the first work addressing the λ-OASMT problem. Compared with most recent works on OARSMT problem, A-OAT obtains up to 30-Kx speedup with quality solution. We have tested randomly generated cases with up to 10 K terminals and 10-K rectilinear obstacles within 4 seconds on a Sun V880 workstation (755-MHz CPU and 4-GB memory). The high efficiency and accuracy of λ-OAT make it extremely practical and useful in the routing phase. [ABSTRACT FROM AUTHOR]