Resolving Braess's Paradox in Random Networks

Braess’s paradox states that removing a part of a network may improve the players’ latency at equilibrium. In this work, we study the approximability of the best subnetwork problem for the class of random Gn,p instances proven prone to Braess’s paradox by Valiant and Roughgarden RSA ’10 (Random Struct Algorithms 37(4):495–515, 2010), Chung and Young WINE ’10 (LNCS 6484:194–208, 2010) and Chung et al. RSA ’12 (Random Struct Algorithms 41(4):451–468, 2012). Our main contribution is a polynomial-time approximation-preserving reduction of the best subnetwork problem for such instances to the corresponding problem in a simplified network where all neighbors of source s and destination t are directly connected by 0 latency edges. Building on this, we consider two cases, either when the total rate r is sufficiently low, or, when r is sufficiently high. In the first case of low r=O(n+), here n+ is the maximum degree of {s,t}, we obtain an approximation scheme that for any constant ε>0 and with high probability, computes a subnetwork and an ε-Nash flow with maximum latency at most (1+ε)L∗+ε, where L∗ is the equilibrium latency of the best subnetwork. Our approximation scheme runs in polynomial time if the random network has average degree O(poly(lnn)) and the traffic rate is O(poly(lnlnn)), and in quasipolynomial time for average degrees up to o(n) and traffic rates of O(poly(lnn)). Finally, in the second case of high r=Ω(n+), we compute in strongly polynomial time a subnetwork and an ε-Nash flow with maximum latency at most (1+2ε+o(1))L∗

Improving selfish routing for risk-averse players

We investigate how and to which extent one can exploit riskaversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy (PoA) is improved. We introduce small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases due to risk-aversion. We adopt the model of γ-modifiable routing games, a variant of routing games with restricted tolls. We prove that computing the best γ-enforceable flow is NP-hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured γ-enforceable flow whose PoA improves fast as γ increases. We show that the complexity of computing such a γ-enforceable flow is determined by the complexity of computing a Nash flow of the original game. Moreover, we prove that the PoA of this flow is best possible in the worst-case, in the sense that there are instances where (i) the best γ-enforceable flow has the same PoA, and (ii) considering more flexible modifications does not lead to any further improvement. © Springer-Verlag Berlin Heidelberg 2015