The Limits of Local Search for Weighted k-Set Packing

Abstract

We consider the weighted k-Set Packing Problem, where, given a collection $\mathcal {S}$ of sets, each of cardinality at most k, and a positive weight function $w:\mathcal {S}\rightarrow \mathbb {R}_{>0}$, the task is to find a sub-collection of $\mathcal {S}$ consisting of pairwise disjoint sets of maximum total weight. As this problem does not permit a polynomial-time $o(\frac{k}{\log k})$-approximation unless $P=NP$ [11], most previous approaches rely on local search. For twenty years, Berman’s algorithm SquareImp [2], which yields a polynomial-time $\frac{k+1}{2}+\epsilon$-approximation for any fixed $\epsilon >0$, has remained unchallenged. Only recently, it could be improved to $\frac{k+1}{2}-\frac{1}{63,700,992}+\epsilon$ by Neuwohner [16]. 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 $\frac{k+1}{3}+\epsilon$. 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 $\frac{k+\epsilon _k}{2}$, where $\lim _{k\rightarrow \infty } \epsilon _k = 0$. On the other hand, we prove that this is asymptotically best possible in that searching for local improvements of logarithmically bounded size cannot produce an approximation ratio below $\frac{k}{2}$.