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 G n , 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 ( ln n ) ) and the traffic rate is O ( poly ( ln ln n ) ) , and in quasipolynomial time for average degrees up to o ( n ) and traffic rates of O ( poly ( ln n ) ) . 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 + 2 ε + o ( 1 ) ) L ∗ .