On the Difficulty of Some Shortest Path Problems

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 = {e(1), e(2), . . . , e(p)} 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 e(i). We show that the replacement paths problem requires Omega(m root n) time in the worst case whenever m = O(n root 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.