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

A classic conjecture of Füredi, Kahn, and Seymour (1993) states that any hypergraph with non‐negative edge weights w(e)$$ w(e) $$ has a matching M$$ M $$ such that ∑e∈M(|e|−1+1/|e|)w(e)≥w∗$$ {\sum}_{e\in M}\left(|e|-1+1/|e|\right)\kern0.3em w(e)\ge {w}^{\ast } $$, where w∗$$ {w}^{\ast } $$ 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∗$$ {\sum}_{e\in M}\left(|e|-\delta (e)\right)\kern0.3em w(e)\ge {w}^{\ast } $$, where δ(e)=|e|/(|e|2+|e|−1)$$ \delta (e)=\mid e\mid /\left({\left|e\right|}^2+|e|-1\right) $$, improving upon the baseline guarantee of ∑e∈M|e|w(e)≥w∗$$ {\sum}_{e\in M}\mid e\mid \kern0.3em w(e)\ge {w}^{\ast } $$.