Some remarks on hypergraph matching and the Füredi-Kahn-Seymour conjecture

11/13/2020

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by   Nikhil Bansal, et al.

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A classic conjecture of Füredi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights w(e), there exists a matching M such that ∑_e ∈ M (|e|-1+1/|e|) w(e) ≥ w^*, where w^* is the value of an optimum fractional matching. We show the conjecture is true for rank-3 hypergraphs, and is achieved by a natural iterated rounding algorithm. While the general conjecture remains open, we give several new improved bounds. In particular, we show that the iterated rounding algorithm gives ∑_e ∈ M (|e|-δ(e)) w(e) ≥ w^*, where δ(e) = |e|/(|e|^2+|e|-1), improving upon the baseline guarantee of ∑_e ∈ M |e| w(e) ≥ w^*.