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 2colorable 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 NPhard 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 2colorable hypergraph on n vertices and dimension d with
O(n11/d
log11/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.
