## Abstract

*R*and

*B*be two disjoint sets of points in the plane such that \(R\cup B\) is in general position, and let \(n=|R\cup B|\). Assume that the points of

*R*are colored red and the points of

*B*are colored blue. A bichromatic plane spanning tree is a plane spanning tree in the complete bipartite geometric graph with bipartition (

*R*,

*B*). In this paper we consider the maximum bichromatic plane spanning tree problem, which is the problem of computing a bichromatic plane spanning tree of maximum total edge length.

- 1.
For the maximum bichromatic plane spanning tree problem, we present an approximation algorithm with ratio 1 / 4 that runs in \(O(n\log n)\) time.

- 2.
We also consider the multicolored version of this problem where the input points are colored with \(k>2\) colors. We present an approximation algorithm that computes a plane spanning tree in a complete

*k*-partite geometric graph, and whose ratio is 1 / 6 if \(k=3\), and 1 / 8 if \(k\geqslant 4\). - 3.
We also revisit the special case of the problem where \(k=n\), i.e., the problem of computing a maximum plane spanning tree in a complete geometric graph. For this problem, we present an approximation algorithm with ratio 0.503; this is an extension of the algorithm presented by Dumitrescu and Tóth (Discrete Comput Geom 44(4):727–752, 2010) whose ratio is 0.502.

- 4.
For points that are in convex position, the maximum bichromatic plane spanning tree problem can be solved in \(O(n^3)\) time. We present an \(O(n^5)\)-time algorithm that solves this problem for the case where the red points lie on a line and the blue points lie on one side of the line.

## Keywords

Multipartite geometric graphs Plane spanning trees Maximum spanning trees Approximation algorithms## Notes

### Acknowledgements

We would like to thank an anonymous referee whose comments improved the readability of the paper. Funding was provide by NSERC and NSF (Grant CCF-1228639)

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