Homology flows, cohomology cuts

We describe the first algorithm to compute maximum flows in surface-embedded graphs in near-linear time. Specifically, given a graph embedded on a surface of genus g, with two specified vertices s and t and integer edge capacities that sum to C, our algorithm computes a maximum (s, ₃^t)₂-flow in O(g⁸n log²n log² C) time. We also present a combinatorial algorithm that takes g^O(g)n^3/2 arithmetic operations. Except for the special case of planar graphs, for which an O(nlogn)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. For graphs of any fixed genus, our algorithms improve these time bounds by roughly a factor of √n. Our key insight is to optimize the homology class of the flow, rather than directly optimizing the flow itself; two flows are in the same homology class if their difference is a weighted sum of directed facial cycles. A dual formulation of our algorithm computes the minimum-cost circulation in a given (real or integer) homology class.