  We consider the weighted $k$-set packing problem, in which we are given a
collection of weighted sets, each with at most $k$ elements and must return a
collection of pairwise disjoint sets with maximum total weight. For $k = 3$,
this problem generalizes the classical 3-dimensional matching problem listed as
one of the Karp's original 21 NP-complete problems. We give an algorithm
attaining an approximation factor of $1.786$ for weighted 3-set packing,
improving on the recent best result of $2-\frac{1}{63,700,992}$ due to
Neuwohner.
  Our algorithm is based on the local search procedure of Berman that attempts
to improve the sum of squared weights rather than the problem's objective. When
using exchanges of size at most $k$, this algorithm attains an approximation
factor of $\frac{k+1}{2}$. Using exchanges of size $k^2(k-1) + k$, we provide a
relatively simple analysis to obtain an approximation factor of 1.811 when $k =
3$. We then show that the tools we develop can be adapted to larger exchanges
of size $2k^2(k-1) + k$ to give an approximation factor of 1.786. Although our
primary focus is on the case $k = 3$, our approach in fact gives slightly
stronger improvements on the factor $\frac{k+1}{2}$ for all $k > 3$. As in
previous works, our guarantees hold also for the more general problem of
finding a maximum weight independent set in a $(k+1)$-claw free graph.
