Single Commodity-Flow Algorithms for Lifts of Graphic and CoGraphic Matroids

Consider a binary matroid $M$ given by its matrix representation. We show that if $M$ is a lift of a graphic or cographic matroid, then in polynomial time (in the size of $M$ and encoding length of the weights) we can either solve the single commodity-flow problem for $M$ or find an obstruction for which the max-flow min-cut relation does not hold. The key tool is an algorithmic version of Lehman's theorem for set covering polyhedra. This tool relies on the ellipsoid method.

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