I have encountered an instance of the Geometric Set Cover problem where the complexity of the union of any subset with size, say k, of m objects is linear with respect to m. I am aware of a well-known approach that utilizes Quasi-Uniform Sampling, originally proposed by Kasturi Varadarajan in STOC 2010. This method, further improved by Timothy M. Chan, leads to a constant approximation factor. In the context of Quasi-Uniform Sampling, Varadarajan introduces a concept known as "union complexity", which requires the union complexity of the objects to be near-linear. In this concept, the complexity of the union of any subset of size k should be near-linear to k. Varadarajan details the use of this concept in the final paragraph of Section 3 of his paper (http://homepage.divms.uiowa.edu/~kvaradar/paps/weightedcover.pdf).
I am reaching out to inquire if there exists a modification to this approach or an alternative method that could be applied to address my specific instance of the problem. I have invested a substantial amount of effort in tackling this challenge and would greatly appreciate any guidance or insights one can provide.