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We consider sel sh routing over a network consisting of m parallel links through which n sel sh users route their tra c trying to minimize their own expected latency. W e study the class of mixed strategies in which the expected latency through each link is at most a constant multiple of the optimum maximum latency had global regulation been available. For the case of uniform links it is known that all Nash equilibria belong to this class of strategies. We a r e i n terested in bounding the coordination ratio (or price of anarchy) of these strategies de ned as the worst-case ratio of the maximum (over all links) expected latency over the optimum maximum latency. The load balancing aspect of the problem immediately implies a lower bound ; ln m ln ln m of the coordination ratio. We g i v e a tight ( u p t o a m ultiplicative c o n s t a n t) upper bound. To show the upper bound, we analyze a variant o f t h e classical balls and bins problem, in which balls with arbitrary weights are placed into bins according to arbitrary probability distributions. At the heart of our approach is a new probabilistic tool that we c a l l

We study the problem of routing traffic through a congested network. We focus on the simplest case of a network consisting of m parallel links. We assume a collection of n network users; each user employs a mixed strategy, which is a probability distribution over links, to control the shipping of its own assigned traffic. Given a capacity for each link specifying the rate at which the link processes traffic, the objective is to route traffic so that the maximum (over all links) latency is minimized. We consider both uniform and arbitrary link capacities.How much decrease in global performance is necessary due to the absence of some central authority to regulate network traffic and implement an optimal assignment of traffic to links? We investigate this fundamental question in the context of Nash equilibria for such a system, where each network user selfishly routes its traffic only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the Coordination Ratio, originally defined by Koutsoupias and Papadimitriou [16], as a measure of the cost of lack of coordination among the users; roughly speaking, the Coordination Ratio is the ratio of the expectation of the maximum (over all links) latency in the worst possible Nash equilibrium, over the least possible maximum latency had global regulation been available.Our chief instrument is a set of combinatorial Minimum Expected Latency Cost Equations, one per user, that characterize the Nash equilibria of this system. These are linear equations in the minimum expected latency costs, involving the user traffics, the link capacities, and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed strategies, where each user assigns its traffic with a strictly positive probability to every link, to derive the first existence and uniqueness results for fully mixed Nash equilibria in this setting. Through a thorough analysis and characterization of fully mixed Nash equilibria, we obtain tight upper bounds of no worse than O(ln n/ln ln n) on the Coordination Ratio for (i) the case of uniform capacities and arbitrary traffics and (ii) the case of arbitrary capacities and identical traffics.

In this work we consider the problem of gathering autonomous robots in the plane. In particular, we consider non-transparent unit-disc robots (i.e., fat) in an asynchronous setting. Vision is the only mean of coordination. Using a state-machine representation we formulate the gathering problem and develop a distributed algorithm that solves the problem for any number of robots.The main idea behind our algorithm is for the robots to reach a configuration in which all the following hold: (a) The robots' centers form a convex hull in which all robots are on the convex, (b) Each robot can see all other robots, and (c) The configuration is connected, that is, every robot touches another robot and all robots together form a connected formation. We show that starting from any initial configuration, the robots, making only local decisions and coordinate by vision, eventually reach such a configuration and terminate, yielding a solution to the gathering problem.

We study the problem of routing tra c through a congested network. We focus on the simplest case of a network consisting of m parallel links. W e assume a collection of n network users, each employing a mixed strategy which is a probability distribution over links, to control the shipping of its own assigned tra c. Given a capacity for each link specifying the rate at which the link processes tra c, the objective i s to route tra c so that the maximum expected latency over all links is minimized. We consider both uniform and nonuniform link capacities.How m uch decrease in global performace is necessary due to the absence of some central authority to regulate network tra c and implement an optimal assignment of tra c to links? We i n vestigate this fundamental question in the context of Nash equilibria for such a system, where each n e t work user sel shly routes its tra c only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the coordination ratio, de ned by Koutsoupias and Papadimitriou 25] as the ratio of the maximum (over all links) expected latency in the worst possible Nash equlibrium, over the least possible maximum latency had global regulation been available, as a measure of the cost of lack of coordination among the network users.Our point of departure is a set of combinatorial minimum expected latency cost equations, one per network user, that characterize the Nash equilibria of this system. These are linear equations in the minimum expected latencies, involving the users' tra cs, the link capacities and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed s t r ategies, where each user assigns its tra c with a non-zero probability t o e v ery link, to derive the rst existence and uniqueness results for Nash equilibria in this setting. Most importantly, we use the derived characterizations of Nash equilibria to show, under the assumption of fully mixed strategies, tight upper bounds of no worse than O(ln n= ln ln n) on the coordination ratio for (i) the case of uniform link capacities and arbitrary tra cs, and (ii) the case of non-uniform link capacities and identical tra cs.