While our algorithm is an adaptation of the native ratio technique m88 movie utilized in earlier works, the evaluation deviates considerably and relies on structural properties of matroid intersection, referred to as kernels. Finally, we also conjecture that our algorithm gives a \((k+\varepsilon )\) approximation for the intersection of k matroids however show that new instruments are needed within the evaluation because the structural properties we use fail for \(k\ge 3\). 2 we introduce primary matroid concepts and we formally outline the weighted matroid intersection drawback in the semi-streaming mannequin.
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T must include b, as it’s the heaviest factor in each matroid. Now, we now have all the lemmas to show our primary theorem which we state under. Both people and organizations that work with arXivLabs have embraced and accepted our values of openness, group, excellence, and user data privacy. ArXiv is dedicated to those values and only works with companions that adhere to them. Watch Jack Draper vs Andrey Rublev within the Qatar Open final reside on Sky Sports Tennis at 3pm on Saturday. Watch the ATP and WTA Tours, in addition to the US Open, stay on Sky Sports Tennis in 2025 or stream with NOW and the Sky Sports App.
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)\) elements at any level if we assume that values of f are polynomially bounded in
While this method is very versatile and it readily generalizes to weighted hypergraph matchings, it is much harder to make use of it for the related drawback of weighted matroid intersection. This is perhaps stunning as lots of the prior outcomes for the matching drawback additionally applies to the matroid intersection problem within the semi-streaming model (see Sect. 2 for definitions). Indeed, the grasping algorithm nonetheless returns a 2-approximate solution in the unweighted case and the algorithm in [5] returns a \((4+\varepsilon )\)-approximate answer for weighted instances. So, previous to our work, the status of the matroid intersection drawback was that of the matching downside earlier than [14]. In this part, we consider the problem of submodular matroid intersection in the semi-streaming model.
)\) and a simple proof shows that it returns a 2-approximate answer in the unweighted case, i.e, a matching of measurement no less than half the size of an maximum matching. However, this primary strategy fails to achieve any approximation guarantee for weighted graphs. Very lately, Levin and Wajc [11] obtained improved approximation ratios for matching and b-matching issues in the semi-streaming model with respect to submodular features.
We now describe at a high-level the reason that the techniques from [14] aren’t easily relevant to matroid intersection and our approach for coping with this issue. This method of setting up the solution M greedily by going backwards in time is a normal framework for analyzing algorithms based mostly on the native ratio approach. Now to be able to adapt their algorithm to matroid intersection, recall that the bipartite matching drawback can be formulated as the intersection of two partition matroids. We can thus reinterpret their algorithm and evaluation on this setting. Furthermore, after this reinterpretation, it’s not too hard to define an algorithm that works for the intersection of any two matroids.
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