Passing the Limits of Pure Local Search for Weighted $k$-Set Packing

We study the weighted $k$-Set Packing problem: Given a collection $S$ of sets, each of cardinality at most $k$, together with a positive weight function $w:\mathcal{S}\rightarrow\mathbb{Q}{>0}$, the task is to compute a disjoint sub-collection $A\subseteq \mathcal{S}$ of maximum total weight. For $k\leq 2$, the weighted $k$-Set Packing problem can be solved in polynomial time, but for $k\geq 3$, it becomes $NP$-hard. Recently, Neuwohner has shown how to obtain approximation guarantees of $\frac{k+\epsilonk}{2}$ with $\lim{k\rightarrow\infty}\epsilonk=0$. She further showed her result to be asymptotically best possible in that no algorithm considering local improvements of logarithmically bounded size with respect to some fixed power of the weight function can yield an approximation guarantee better than $\frac{k}{2}$. In this paper, we finally show how to beat the threshold of $\frac{k}{2}$ for the weighted $k$-Set Packing problem by $\Omega(k)$. We achieve this by combining local search with the application of a black box algorithm for the unweighted $k$-Set Packing problem to carefully chosen sub-instances. In doing so, we manage to link the approximation ratio for general weights to the one achievable in the unweighted case and we obtain guarantees of at most $\frac{k+1}{2}-2\cdot 10^{-4}$ for all $k\geq 4$.