Improved algorithms for min cut and max flow in undirected planar graphs

Abstract

We study the min st-cut and max st-flow problems in planar graphs, both in static and in dynamic settings. First, we present an algorithm that given an undirected planar graph and two vertices s and t computes a min st-cut in O(n log log n) time. Second, we show how to achieve the same bound for the problem of computing a max st-flow in an undirected planar graph. These are the first algorithms breaking the O(n log n) barrier for those two problems, which has been standing for more than 25 years. Third, we present a fully dynamic algorithm maintaining the value of the min st-cuts and the max st-flows in an undirected plane graph (i.e., a planar graph with a fixed embedding): our algorithm is able to insert and delete edges and answer queries for min st-cut/max st-flow values between any pair of vertices s and t in O(n(2/3) log(8/3) n) time per operation. This result is based on a new dynamic shortest path algorithm for planar graphs which may be of independent interest. We remark that this is the first known non-trivial dynamic algorithm for min st-cut and max st-flow.