Shadow Prices vs. Vickrey Prices in Multipath Routing

Shadow price and Vickrey price are two classic metrics that can be applied to measure the relative importance of links in a communication network. Each metric has been extensively investigated and enjoys important applications. We study the underlying connections between these two metrics with seemingly different definitions, under a general mathematical model of multipath multi-session multicast routing. We show that Vickrey prices provide upper-bounds for shadow prices in general, and the fine granularity version of Vickrey price, unit Vickrey price, equals exactly the maximum shadow price. We further design an efficient algorithm that computes all-link max/min shadow prices and unit Vickrey prices simultaneously, for unicast routing, reducing the complexity of a straightforward algorithm by an order of O(|E|).