Escaping Braess's paradox through approximate Caratheodory's theorem

In selfish routing, Braess's paradox demonstrates the counterintuitive fact that removing a part of a network may improve the players' cost at equilibrium. In this work, we use the approximate version of Caratheodory's theorem (Barman, SICOMP 2018) and show how to efficiently approximate the best subnetwork on selfish routing instances with relatively short paths and Lipschitz continuous cost functions. Restriction to networks with short paths is necessary to escape the strong inapproximability result of (Roughgarden, JCSS 2006). We present the first polynomial-time algorithm that approximates (in a certain relaxed sense) the best subnetwork problem on such instances. Specifically, for any constant epsilon > 0, our algorithm computes an epsilon-Nash flow with maximum latency at most (1 + epsilon)L* + epsilon, where L* is the equilibrium latency of the best subnetwork. Moreover, our algorithm runs in quasipolynomial time on networks with polylogarithmically long paths. As a corollary, we obtain the first polynomial-time approximation scheme for the best subnetwork in the class of random Gn,p instances proven prone to Braess's paradox by (Roughgarden and Valiant, RSA 2010) and (Chung Graham et al., RSA 2012). (c) 2022 Elsevier B.V. All rights reserved.