  A classic conjecture of F\"{u}redi, Kahn and Seymour (1993) states that given
any hypergraph with non-negative edge weights $w(e)$, there exists a matching
$M$ such that $\sum_{e \in M} (|e|-1+1/|e|)\, w(e) \geq w^*$, 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 $\sum_{e \in
M} (|e|-\delta(e))\, w(e) \geq w^*$, where $\delta(e) = |e|/(|e|^2+|e|-1)$,
improving upon the baseline guarantee of $\sum_{e \in M} |e|\,w(e) \geq w^*$.
