Convex hulls, oracles, and homology☆

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Abstract

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”-case of CompletenessC has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).

Keywords

Convex hull algorithm
Simplicial homology computation

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An extended abstract version of this paper, “Polytope verification by homology computation”, has appeared in the Proceedings of EuroCG, Berlin, March 26–28, 2001, pp. 142–145.

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