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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. 

The best known solutions for k-message broadcast in dynamic networks of size n require Ω(nk) rounds. In this paper, we see if these bounds can be improved by smoothed analysis. To do so, we study perhaps the most natural randomized algorithm for disseminating tokens in this setting: at every time step, choose a token to broadcast randomly from the set of tokens you know. We show that with even a small amount of smoothing (i.e., one random edge added per round), this natural strategy solves k-message broadcast in Õ(n+k³) rounds, with high probability, beating the best known bounds for k = o(√n) and matching the Ω(n+k) lower bound for static networks for k = O(n^{1/3}) (ignoring logarithmic factors). In fact, the main result we show is even stronger and more general: given 𝓁-smoothing (i.e., 𝓁 random edges added per round), this simple strategy terminates in O(kn^{2/3}log^{1/3}(n)𝓁^{-1/3}) rounds. We then prove this analysis close to tight with an almost-matching lower bound. To better understand the impact of smoothing on information spreading, we next turn our attention to static networks, proving a tight bound of Õ(k√n) rounds to solve k-message broadcast, which is better than what our strategy can achieve in the dynamic setting. This confirms the intuition that although smoothed analysis reduces the difficulties induced by changing graph structures, it does not eliminate them altogether. Finally, we apply tools developed to support our smoothed analysis to prove an optimal result for k-message broadcast in so-called well-mixed networks in the absence of smoothing. By comparing this result to an existing lower bound for well-mixed networks, we establish a formal separation between oblivious and strongly adaptive adversaries with respect to well-mixed token spreading, partially resolving an open question on the impact of adversary strength on the k-message broadcast problem.