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While this system is very versatile and it readily generalizes to weighted hypergraph matchings, it’s a lot harder to make use of it for the associated drawback of weighted matroid intersection. This is maybe shocking as lots of the prior results for the matching downside additionally applies to the matroid intersection downside within the semi-streaming model (see Sect. 2 for definitions). Indeed, the greedy algorithm still returns a 2-approximate resolution within the unweighted case and the algorithm in [5] returns a \((4+\varepsilon )\)-approximate resolution for weighted situations. So, previous to our work, the status of the matroid intersection drawback was that of the matching drawback earlier than [14]. In this section, we consider the issue of submodular matroid intersection within the semi-streaming model.
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However, bipartite matching is a particular case of matroid intersection which captures a wealthy set of seemingly extra complicated issues. This added expressiveness causes the evaluation and the standard framework for analyzing native ratio algorithms to fail. Specifically, we prove that an answer shaped by operating the greedy algorithm on S within the reverse order (as carried out for the matching problem) fails to offer any constant-factor approximation guarantee for the matroid intersection problem. While the fundamental grasping algorithm offers a semi-streaming algorithm with an approximation assure of two for the unweighted matching drawback, it was only lately that Paz and Schwartzman obtained a similar end result for weighted cases. Their approach is based on the versatile local ratio method and in addition applies to generalizations such as weighted hypergraph matchings. Our methods additionally permit us to generalize current results by Levin and Wajc on submodular maximization topic to matching constraints to that of matroid-intersection constraints.
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