Abstract
A coloring of a hypergraph is a mapping of vertices to colors such that no hyperedge is monochromatic. We are interested in the problem of coloring 2-colorable hypergraphs. For the special case of graphs (hypergraphs of dimension 2) this can easily be done in linear time. The problem for general hypergraphs is much more difficult since a result of Lovasz implies that the problem is NP-hard even if all hyperedges have size three.
In this paper we develop approximation algorithms for this problem. Our first result is an algorithm that colors any 2-colorable hypergraph on n vertices and dimension d with O(n 1−1/d log1−1/d n) colors. This is the first algorithm that achieves a sublinear number of colors in polynomial time. This algorithm is based on a new technique for reducing degrees in a hypergraph that should be of independent interest. For the special case of hypergraphs of dimension three we improve on the previous result by obtaining an algorithm that uses only O(n 2/9 log17/8 n) colors. This result makes essential use of semidefinite programming. This last result is rather surprising because we show that semidefinite programming will fail for any larger dimension.
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© 1996 Springer-Verlag Berlin Heidelberg
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Kelsen, P., Mahajan, S., Ramesh, H. (1996). Approximate hypergraph coloring. In: Karlsson, R., Lingas, A. (eds) Algorithm Theory — SWAT'96. SWAT 1996. Lecture Notes in Computer Science, vol 1097. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61422-2_119
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DOI: https://doi.org/10.1007/3-540-61422-2_119
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