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Hi-index | 5.23 |
Let M be a single s-t network of parallellinks with load dependent latency functions shared by an infinitenumber of selfish users. This may yield a Nash equilibriumwith unbounded Coordination Ratio [E. Koutsoupias, C.Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium onTheoretical Aspects of Computer Science, STACS, vol. 1563, 1999,pp. 404-413; T. Roughgarden, E. Tardos, How bad is selfish routing?in: 41st IEEE Annual Symposium of Foundations of Computer Science,FOCS, 2000, pp. 93-102]. A Leader can decrease thecoordination ratio by assigning flow αr on M,and then all Followers assign selfishly the(1-α)r remaining flow. This is a StackelbergScheduling Instance(M,r,α),0≤α≤1. Itwas shown [T. Roughgarden, Stackelberg scheduling strategies, in:33rd Annual Symposium on Theory of Computing, STOC, 2001, pp.104-113] that it is weakly NP-hard to compute the optimal Leader'sstrategy. For any such network M we efficiently compute the minimumportion @b"M of flow r0 needed by a Leader to induce M'soptimum cost, as well as her optimal strategy. This shows that theoptimal Leader's strategy on instances (M,r,@a=@b"M) is in P.Unfortunately, Stackelberg routing in more general nets can bearbitrarily hard. Roughgarden presented a modification ofBraess's Paradox graph, such that no strategycontrolling αr flow can induce ≤1/α times theoptimum cost. However, we show that our main result alsoapplies to any s-t net G. We take careof the Braess's graph explicitly, as a convincing example. Finally,we extend this result to k commodities. A conference versionof this paper has appeared in [A. Kaporis, P. Spirakis, The priceof optimum in stackelberg games on arbitrary single commoditynetworks and latency functions, in: 18th annual ACM symposium onParallelism in Algorithms and Architectures, SPAA, 2006, pp.19-28]. Some preliminary results have also appeared as technicalreport in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The priceof optimum in stackelberg games, in: Electronic Colloquium onComputational Complexity, ECCC, (056), 2005].