Stronger Bounds on Braess's Paradox and the Maximum Latency of Selfish Routing

We give several new upper and lower bounds on the worst-case severity of Braess's paradox and the price of anarchy of selfish routing with respect to the maximum latency objective. In single-commodity networks with arbitrary continuous and nondecreasing latency functions, we prove that this worst-case price of anarchy is exactly $n-1$, where $n$ is the number of network vertices. For Braess's paradox in such networks, we prove that removing at most $c$ edges from a network decreases the common latency incurred by traffic at equilibrium by at most a factor of $c+1$. In particular, the worst-case severity of Braess's paradox with a single edge removal is maximized in Braess's original four-vertex network. In multicommodity networks, we exhibit an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both the worst-case severity of Braess's paradox and the price of anarchy for the maximum latency objective grow exponentially with the network size. This construction demonstrates that numerous known selfish routing results for single-commodity networks have no analogues in networks with two or more commodities. We also prove an upper bound on both of these quantities that is exponential in the network size and independent of the network latency functions, showing that our construction is close to optimal. Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability.