Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs (Adrian Vladu)
Channel: IRIF
Date: 2021-04-20

We present an m^{4/3+o(1)} log W -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite b-matching when |b|_1 = O(m), and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities, due to Liu and Sidford (FOCS, 2020).
Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around some carefully chosen circulations.

This talk will be self-contained and will assume no prior background in optimization. Based on https://arxiv.org/abs/2003.04863, appears in FOCS 2020.