Tight Bound on Vertex Cut Sparsifiers in Directed Acyclic Graphs
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For an unweighted graph on k terminals, Kratsch and Wahlström constructed a vertex sparsifier with O(k^3) vertices via the theory of representative families on matroids. Since their breakthrough result in 2012, no improvement upon the O(k^3) bound has been found. In this paper, we interpret Kratsch and Wahlström's result through the lens of Bollobás's Two-Families Theorem from extremal combinatorics. This new perspective allows us to close the gap for directed acyclic graphs and obtain a tight bound of Θ(k^2). Central to our approach is the concept of skew-symmetry from extremal combinatorics, and we derive a similar theory for the representation of skew-symmetric families that may have future applications.
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