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In decentralized optimization over networks, synchronizing the updates of all nodes incurs significant communication overhead. For this reason, much of the recent literature has focused on the analysis and design of asynchronous optimization algorithms where nodes can activate anytime and contact a single neighbor to complete an iteration together. However, most works assume that the neighbor selection is done at random based on a fixed probability distribution (eg, uniform), a choice that ignores the optimization landscape at the moment of activation. Instead, in this work we introduce an optimization-aware selection rule that chooses the neighbor providing the highest dual cost improvement (a quantity related to a dualization of the problem based on consensus). This scheme is related to the coordinate descent (CD) method with the Gauss-Southwell (GS) rule for coordinate updates; in our setting however, only a subset of coordinates is accessible at each iteration (because each node can communicate only with its neighbors), so the existing literature on GS methods does not apply. To overcome this difficulty, we develop a new analytical framework for smooth and strongly convex functions that covers our new class of set-wise CD algorithms–a class that applies to both decentralized and parallel distributed computing scenarios–and we show that the proposed set-wise GS rule can speed up the convergence in terms of iterations by a factor equal to the size of the largest coordinate set. We analyze extensions of these algorithms that exploit the knowledge of the smoothness constants when available and otherwise propose an algorithm to …