Hanno Lefmann

The algorithmic aspects of the regularity lemma

Abstract The regularity lemma of Szemeredi asserts that every graph can be partitioned 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 co-NP-complete. 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 n -vertex graph, can be found in time O ( M ( n )), where M ( n ) = O ( n 2.376 ) is the time needed to multiply two n by n matrices with 0, 1-entries over the integers. The algorithm can be parallelized and implemented in NC 1 . 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 P and NP differ), our constructive version of the regularity lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.

A Fast Approximation Algorithm for Computing theFrequencies of Subgraphs in a Given Graph

In this paper we give an algorithm which, given a labeled graph on

On uncrowded hypergraphs

n

On canonical Ramsey numbers for complete graphs versus paths

vertices and a list of all labeled graphs on

On Heilbronn's problem in higher dimension

k

The Algorithmic Aspects of Uncrowded Hypergraphs

vertices, provides for each graph

Infinite partition regular matrices

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Monochromatic and zero-sum sets of nondecreasing diameter

of this list an approximation to the number of induced copies of

Multicolored Subsets in Colored Hypergraphs

H

Point sets with distinct distances

in