Erratum to "Vickrey Pricing and Shortest Paths: What is an Edge Worth?"

A naı̈ve algorithm for the replacement paths problem runs in O(n(m+n logn)) time, executing the single-source shortest path algorithm up to n times. In our paper [3], we claimed that the replacement paths problem can be solved inO(m+n logn) time for both undirected and directed graphs. However, there is a flaw that invalidates the algorithm for directed graphs. (The algorithm for undirected graphs remains valid.) The same bound for undirected graphs is also achieved by Nardelli, Proietti, and Widmayer [6], who solve the most vital node problem with an algorithm that also solves the replacement paths problem. Their work is in turn based on earlier work by Malik, Mittal and Gupta [5], Ball, Golden, and Vohra [1] and BarNoy, Khuller, and Schieber [2]. The error in our algorithm for directed graphs led us to investigate the hardness of the replacement paths problem and other related problems. We have recently established lower bounds that show that no replacement paths algorithm of a certain class (including all known algorithms) can achieve the running time of the algorithm for undirected graphs [4]. The mistake in the directed graph algorithm of [3] occurs on page 257, just after Lemma 2, where we say “A simple corollary of this lemma is the fact that if (u; v) is the single edge of path(x; y; G n e) in E(Vx; Vy), then