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While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints that might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the 𝓁_p-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the 𝓁_∞-norm) and the number of edges (the 𝓁₁-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogarithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Guy Kortsarz and Zeev Nutov, 2022] and [X. Guo et al., 2022]. 

The SetCover problem has been extensively studied in many different models of computation, including parallel and distributed settings. From an approximation point of view, there are two standard guarantees: an O(log Δ)-approximation (where Δ is the maximum set size) and an O(f)-approximation (where f is the maximum number of sets containing any given element). In this paper, we introduce a new, surprisingly simple, model-independent approach to solving SetCover in unweighted graphs. We obtain multiple improved algorithms in the MPC and CRCW PRAM models. First, in the MPC model with sublinear space per machine, our algorithms can compute an O(f) approximation to SetCover in Ô(√{log Δ} + log f) rounds and a O(log Δ) approximation in O(log^{3/2} n) rounds. Moreover, in the PRAM model, we give a O(f) approximate algorithm using linear work and O(log n) depth. All these bounds improve the existing round complexity/depth bounds by a log^{Ω(1)} n factor. Moreover, our approach leads to many other new algorithms, including improved algorithms for the HypergraphMatching problem in the MPC model, as well as simpler SetCover algorithms that match the existing bounds. 

A t-emulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant t-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior. In particular, our main result is that, for (2k-1)-emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any n-node input graph, we construct a 5-emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that Ω(n^{4/3}) edges are necessary even if the 5-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k.