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We consider routing games where the performance of each user is dictated by the worst (bottleneck) element it employs. We are given a network, finitely many (selfish) users, each associated with a positive flow demand, and a load-dependent performance function for each network element; the social (i.e., system) objective is to optimize the performance of the worst element in the network (i.e., the network bottleneck). Although we show that such "bottleneck" routing games appear in a variety of practical scenarios, they have not been considered yet. Accordingly, we study their properties, considering two routing scenarios, namely when a user can split its traffic over more than one path (splittable bottleneck game) and when it cannot (unsplittable bottleneck game). First, we prove that, for both splittable and unsplittable bottleneck games, there is a (not necessarily unique) Nash equilibrium. Then, we consider the rate of convergence to a Nash equilibrium in each game. Finally, we investigate the efficiency of the Nash equilibria in both games with respect to the social optimum; specifically, while for both games we show that the price of anarchy is unbounded, we identify for each game conditions under which Nash equilibria are socially optimal.

Abstract-Unlike traditional routing schemes that route all traffic along a single path, multipath routing strategies split the traffic among several paths in order to ease congestion. It has been widely recognized that multipath routing can be fundamentally more efficient than the traditional approach of routing along single paths. Yet, in contrast to the single-path routing approach, most studies in the context of multipath routing focused on heuristic methods. We demonstrate the significant advantage of optimal (or near optimal) solutions. Hence, we investigate multipath routing adopting a rigorous (theoretical) approach. We formalize problems that incorporate two major requirements of multipath routing. Then, we establish the intractability of these problems in terms of computational complexity. Finally, we establish efficient solutions with proven performance guarantees.

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Neural network quantization enables the deployment of large models on resource-constrained devices. Current post-training quantization methods fall short in terms of accuracy for INT4 (or lower) but provide reasonable accuracy for INT8 (or above). In this work, we study the effect of quantization on the structure of the loss landscape. We show that the structure is flat and separable for mild quantization, enabling straightforward post-training quantization methods to achieve good results. We show that with more aggressive quantization, the loss landscape becomes highly non-separable with steep curvature, making the selection of quantization parameters more challenging. Armed with this understanding, we design a method that quantizes the layer parameters jointly, enabling significant accuracy improvement over current post-training quantization methods. Reference implementation is available at https:// github. com/ ynahs han/ nn-quant izati on-pytor ch/ tree/ master/ lapq.