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