Note – Uncrossing a Family of Set-Pairs

Frank and Jordán [1] proved an important min-max result on covering a crossing family of set-pairs. As an application, among others they can solve the unweighted node-connectivity augmentation problem for directed graphs in polynomial time. In this paper, we show how to solve the dual packing problem in polynomial time. To decompose a fractional dual optimum as a convex combination of integer vertices, besides the ellipsoid method, we use a polynomial-time algorithm for uncrossing a family of set-pairs. Our main result is this uncrossing algorithm.