On the difficulty of some shortest path problems

TLDR: In this paper, a superlinear lower bound of Ω(m √n) was shown for the replacement path problem in directed graphs, where m = O(n √ n).

Abstract: We prove superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = le1, e2, …, epr between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges ei. We show that the replacement paths problem requires Ω(m √n) time in the worst case whenever m = O(n √n). Our construction also implies a similar lower bound on the k shortest simple paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest simple paths) can be solved in near-linear time for undirected graphs.