  A well-known conjecture of F\"uredi, Kahn, and Seymour (1993) on non-uniform
hypergraph matching states that for any hypergraph with edge weights $w$, there
exists a matching $M$ such that the inequality $\sum_{e\in M} g(e) w(e) \geq
\mathrm{OPT}_{\mathrm{LP}}$ holds with $g(e)=|e|-1+\frac{1}{|e|}$, where
$\mathrm{OPT}_{\mathrm{LP}}$ denotes the optimal value of the canonical LP
relaxation.
  While the conjecture remains open, the strongest result towards it was very
recently obtained by Brubach, Sankararaman, Srinivasan, and Xu
(2020)---building on and strengthening prior work by Bansal, Gupta, Li, Mestre,
Nagarajan, and Rudra (2012)---showing that the aforementioned inequality holds
with $g(e)=|e|+O(|e|\exp(-|e|))$.
  Actually, their method works in a more general sampling setting, where, given
a point $x$ of the canonical LP relaxation, the task is to efficiently sample a
matching $M$ containing each edge $e$ with probability at least
$\frac{x(e)}{g(e)}$.
  We present simpler and easy-to-analyze procedures leading to improved
results. More precisely, for any solution $x$ to the canonical LP, we introduce
a simple algorithm based on exponential clocks for Brubach et al.'s sampling
setting achieving $g(e)=|e|-(|e|-1)x(e)$.
  Apart from the slight improvement in $g$, our technique may open up new ways
to attack the original conjecture.
  Moreover, we provide a short and arguably elegant analysis showing that a
natural greedy approach for the original setting of the conjecture shows the
inequality for the same $g(e)=|e|-(|e|-1)x(e)$ even for the more general
hypergraph $b$-matching problem.
