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 $k\geq 3$.

翻译：我们把最大重量独立设置问题( MWIS) 放在 $+1 的免费图表中。 也就是说, 在给定的 $- claw 免费图形中计算一组最高重量 $G= (V,E), 配有正重的重量函数 $w: V\right\ mathbb{R ⁇ 0}。 对于 $\ geqq 美元 1, $+1 的 MWIS 免费图表将加权的 $+1 美元- set包装问题概括化。 鉴于 $k\ ge- set 3$, 这个问题不允许在给给给定的 $droupal $(美元- coloria 美元 美元 美元) 上计算一个独立的最高重量。 在前二十年, Berman' sal commax squal compatial 3- compax$(美元), 在给IMIS IMFI 上显示 问题是如何在为 美元。