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Summary: Approximate Hypergraph Coloring
Noga Alon 1 Pierre Kelsen 2 Sanjeev Mahajan 3 Hariharan Ramesh 4
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 Lov´asz 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(n1-1/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(n2/9
log
17
8 n) colors. This result makes essential use of semidefinite programming.
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