Resolving Braess’s Paradox in Random Networks

Abstract

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