Abstract
Drawing a graph in a way that minimizes the number of edge-crossings is a well-studied problem. Recently there has been work characterizing both the minimum and maximum number of edge-crossings possible in various graph classes, assuming rectilinear (straight-line) edges. In this paper, we investigate the algorithmic problem of maximizing the number of edge-crossings over all rectilinear drawings a graph. We show that this problem is NP-hard and lies in \(\exists \mathbb {R}\). We give a nontrivial derandomization of the natural randomized 1/3-approximation algorithm, which generalizes to a weighted setting as well as to an ordering constraint satisfaction problem. We evaluate these algorithms and other heuristics in simulation.
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For ease of exposition, we refer simply to the number of crossings for the remainder of the proof, although exactly the same analysis applies in the weighted setting, whether the weights of crossings are based on underlying edge weights or are permitted to be completely arbitrary, as in an OCSP.
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Acknowledgements
This work was supported in part by PSC-CUNY Research Award 67665-00 45, CUNY Collaborative Incentive Research Grant (CIRG 21) 2153, and a Research in the Classroom Idea Grant.
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Bald, S., Johnson, M.P., Liu, O. (2016). Approximating the Maximum Rectilinear Crossing Number. In: Dinh, T., Thai, M. (eds) Computing and Combinatorics . COCOON 2016. Lecture Notes in Computer Science(), vol 9797. Springer, Cham. https://doi.org/10.1007/978-3-319-42634-1_37
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