Abstract
We study the bipartite crossing number problem. When the minimum degree and the maximum degree of the graph are close to each other, we derive two polynomial time approximation algorithms for solving this problem, with approximation factors, O(log2 n), and O(log n log log n), from the optimal, respectively, where n is the number of vertices. This problem had been known to be NP-hard, but no approximation algorithm which could generate a provably good solution was known. An important aspect of our work has been relating this problem to the linear arrangement problem. Indeed using this relationship we also present an O(n 1.6) time algorithm for computing the bipartite crossing number of a tree.
We also settle down the problem of computing a largest weighted biplanar subgraph of an acyclic graph by providing a linear time algorithm to it. This problem was known to be NP-hard when graph is planar and very sparse, and all weights are 1.
The research of the first author was supported by NSF grant CCR-9528228.
Research of the 2nd and the 4th author was partially supported by grant No. 95/5305/95 of Slovak Grant Agency and Alexander von Humboldt Foundation.
Research of the third author was supported in part by the Hungarian National Science Fund contracts T 016 358 and T 019 367.
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© 1997 Springer-Verlag Berlin Heidelberg
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Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt'o, I. (1997). On bipartite crossings, largest biplanar subgraphs, and the linear arrangement problem. In: Dehne, F., Rau-Chaplin, A., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 1997. Lecture Notes in Computer Science, vol 1272. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63307-3_48
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DOI: https://doi.org/10.1007/3-540-63307-3_48
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