  We consider the Maximum Weight Independent Set Problem (MWIS) in $d$-claw
free graphs, i.e. the task of computing an independent set of maximum weight in
a given $d$-claw free graph $G=(V,E)$ equipped with a positive weight function
$w:V\rightarrow\mathbb{R}_{>0}$. For $k\geq 1$, the MWIS in $k+1$-claw free
graphs generalizes the weighted $k$-Set Packing Problem. Given that for $k\geq
3$, this problem does not permit a polynomial time $o(\frac{k}{\log
k})$-approximation unless $P=NP$, most previous algorithms for both weighted
$k$-Set Packing and the MWIS in $d$-claw free graphs rely on local search. For
the last twenty years, Berman's algorithm SquareImp, which yields a
$\frac{d}{2}+\epsilon$-approximation for the MWIS in $d$-claw free graphs, has
remained unchallenged for both problems. Recently, it was improved by
Neuwohner, obtaining an approximation guarantee slightly below $\frac{d}{2}$,
and inevitably raising the question of how far one can get by using local
search. In this paper, we finally answer this question asymptotically in the
following sense: By considering local improvements of logarithmic size, we
obtain approximation ratios of $\frac{d-1+\epsilon_d}{2}$ for the MWIS in
$d$-claw free graphs for $d\geq 3$ in quasi-polynomial time, where $0\leq
\epsilon_d\leq 1$ and $\lim_{d\rightarrow\infty}\epsilon_d = 0$. By employing
the color coding technique, we can use the previous result to obtain a
polynomial time $\frac{k+\epsilon_{k+1}}{2}$-approximation for weighted $k$-Set
Packing. On the other hand, we provide examples showing that no local
improvement algorithm considering local improvements of size
$\mathcal{O}(\log(|\mathcal{S}|))$ with respect to some power $w^\alpha$ of the
weight function, where $\alpha\in\mathbb{R}$ is chosen arbitrarily, but fixed,
can yield an approximation guarantee better than $\frac{k}{2}$ for the weighted
$k$-Set Packing Problem with $k\geq 3$.
