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 ${\mathcal{G}}_{n,p}$ 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 $\mathit{ϵ}>0$ and with high probability, computes a subnetwork and an $\mathit{ϵ}$ ‐Nash flow with maximum latency at most $(1+\mathit{ϵ})L^{\ast }+\mathit{ϵ}$ , where $L^{\ast }$ is the equilibrium latency of the best subnetwork. Our approximation scheme runs in polynomial time if the random network has average degree $O(\mathrm{poly}(lnn))$ and the traffic rate is $O(\mathrm{poly}(lnlnn))$ , and in quasipolynomial time for average degrees up to o(n) and traffic rates of $O(\mathrm{poly}(lnn))$ . Finally, in the second case of high $r=\mathit{\Omega }(n_{+})$ , we compute in strongly polynomial time a subnetwork and an $\mathit{ϵ}$ ‐Nash flow with maximum latency at most $(1+2\mathit{ϵ}+o(1))L^{\ast }$ .