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

Abstract

A classic conjecture of Füredi, Kahn, and Seymour (1993) states that any hypergraph with non-negative edge weights w(e) has 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 * .