The limits of local search for weighted k-set packing

Abstract

AbstractWe consider the weighted k-set packing problem, where, given a collection $${\mathcal {S}}$$ S of sets, each of cardinality at most k, and a positive weight function $$w:{\mathcal {S}}\rightarrow {\mathbb {Q}}_{>0}$$ w : S → Q > 0 , the task is to find a sub-collection of $${\mathcal {S}}$$ S consisting of pairwise disjoint sets of maximum total weight. As this problem does not permit a polynomial-time $$o(\frac{k}{\log k})$$ o ( k log k ) -approximation unless $$P=NP$$ 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 $$\frac{k+1}{2}+\epsilon $$ k + 1 2 + ϵ -approximation for any fixed $$\epsilon >0$$ ϵ > 0 , has remained unchallenged. Only recently, it could be improved to $$\frac{k+1}{2}-\frac{1}{63,700,993}$$ 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 $$\frac{k+1}{3}+\epsilon $$ k + 1 3 + ϵ (Fürer 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 $$\frac{k+\epsilon _k}{2}$$ k + ϵ k 2 , where $$\lim _{k\rightarrow \infty } \epsilon _k = 0$$ lim k → ∞ ϵ 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 $$\frac{k}{2}$$ k 2 .