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The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1 / 2 -approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work Konrad and Naidu (Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2021, August 16–18, 2021. LIPIcs, vol 207, pp 19:1–19:18, 2021. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.19) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1 / 2 -approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms were previously known, and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs. The main significance of our results is not in the numerical improvements, but in demonstrating the potential of non-maximal-matching-first algorithms. [ABSTRACT FROM AUTHOR]

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%5f19), which yields a polynomial-time k + 1 2 + ϵ -approximation for any fixed ϵ > 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 + ϵ (Fürer and Yu in International symposium on combinatorial optimization, Springer, 2014. https://doi.org/10.1007/978-3-319-09174-7%5f35). 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 + ϵ k 2 , where 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 k 2 . [ABSTRACT FROM AUTHOR]