The Limits of Local Search for the Maximum Weight Independent Set Problem in d-Claw Free Graphs

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 $\frac{d-1+\epsilond}{2}$ for the MWIS in $d$-claw free graphs for $d\geq 3$ in quasi-polynomial time, where $0\leq \epsilond\leq 1$ and $\lim{d\rightarrow\infty}\epsilond = 0$. 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 $k\geq 3$.