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Abstract : |
We continue the study initiated in [Ro01] on Stackelberg Scheduling Strategies. We are given a set of independent parallel machines or equivalently a set of parallel edges on which certain flow has to be sent. Each edge is endowed with a latency function ?. The setting is that of a non-cooperative game: players choose edges so as minimize their individual latencies. Additionally, there is a single player who control as fraction ? of the total flow. The goal is to find a strategy for the leader (i.e. an assignment of flow to indivual links) such that the selfish users react so as to minimize the total latency of the system. Building on the recent results in [Ro01, RT00], we show the following: 1. We devise a fully polynomial approximate Stackelberg scheme: given a performance ? requirement, the stackelberg scheme runs in time polynomial in and and produces an assignment of flows such that the cost of the induced Nash equilibrium is within a ? factor of the optimum stackelberg ? ? strategy. The result is extended to obtain a polynomial-approximation scheme when instances are restricted to layered directed graphs in which each layer has a bounded number of vertices. 2. We then consider a two round Stackelberg strategy (denoted 2SS). In this strategy, the game consists of three rounds: a move by the leader followed by the moves of all the followers folowed again by a move by the leader who possibly reassigns some of the flows. We show that 2SS always dominates the one round scheme, and for some classes of latency functions, is guaranteed to be closer to the global social optimum. We also consider the variant where the leader plays after the selfish users have routed themselves, and observe that this dominates the one-round scheme. Extensions of the results to the special case when all the latency functions are linear are also presented. Our results extend the earlier results and answer an open question posed by Roughgarden [Ro01]., |
