The limits of local search for weighted k-set packing

We consider the weighted k-set packing problem, where, given a collection S of sets, each of cardinality at most k, and a positive weight function w : S -> Q(>0), the task is to find a sub-collection of S consisting of pairwise disjoint sets of maximum total weight. As this problem does not permit a polynomial-time o( k/ log k )-approximation unless P = N P (Hazan et al. in Comput Complex 15:20-39, 2006. https://doi.org/ 10.1007/s00037-006-0205-6), most previous approaches rely on local search. For twenty years, Berman's algorithm SquareImp (Berman, in: Scandinavian workshop on algorithm theory, Springer, 2000. https://doi.org/10.1007/3-540-44985-X_19), which yields a polynomial-time k+1/ 2 + epsilon-approximation for any fixed E > 0, has remained unchallenged. Only recently, it could be improved to k+1/2 - 1/63,700,993 by Neuwohner (38th International symposium on theoretical aspects of computer science (STACS 2021), Leibniz international proceedings in informatics (LIPIcs), 2021. https://doi.org/ 10.4230/LIPIcs.STACS.2021.53). In her paper, she showed that instances for which the analysis of SquareImp is almost tight are "close to unweighted" in a certain sense. But for the unit weight variant, the best known approximation guarantee is k+1 /3 + is an element of (Furer and Yu in International symposium on combinatorial optimization, Springer, 2014. https://doi.org/10.1007/978-3-319-09174-7_35). Using this observation as a starting point, we conduct a more in-depth analysis of close-to-tight instances of SquareImp. This finally allows us to generalize techniques used in the unweighted case to the weighted setting. In doing so, we obtain approximation guarantees of k+is an element of k/ 2 , where lim(k ->infinity) is an element of(k) = 0. On the other hand, we prove that this is asymptotically best possible in that local improvements of logarithmically bounded size cannot produce an approximation ratio below k/2.