 
Summary: The Algorithmic Aspects of the Regularity Lemma
N. Alon
R. A. Duke
H. Lefmann §
V. Ršodl ¶
R. Yuster
Abstract
The Regularity Lemma of SzemerŽedi is a result that asserts that every graph can be par
titioned in a certain regular way. This result has numerous applications, but its known proof
is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular
partition; we show that deciding if a given partition of an input graph satisfies the properties
guaranteed by the lemma is coNPcomplete. However, we also prove that despite this difficulty
the lemma can be made constructive; we show how to obtain, for any input graph, a partition
with the properties guaranteed by the lemma, efficiently. The desired partition, for an nvertex
graph, can be found in time O(M(n)), where M(n) = O(n2.376
) is the time needed to multiply
two n by n matrices with 0, 1entries over the integers. The algorithm can be parallelized and
implemented in NC1
. Besides the curious phenomenon of exhibiting a natural problem in which
the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if
