An Algorithmic Hypergraph Regularity Lemma

Szemer´edi’s Regularity Lemma [22, 23] is one of the most powerful tools in combinatorics. It asserts that all large graphs G admit a bounded partition of E(G), most classes of which consist of regularly distributed edges. The original proof of this result was non-constructive. A constructive proof was given by Alon, Duke, Lefmann, R¨odl and Yuster [1], which allows one to efficiently construct a regular partition for any large graph. Szemer´edi’s Regularity Lemma was extended to hypergraphs by various authors. Frankl and R¨odl [3] gave one such extension to 3-uniform hypergraphs, and R¨odl and Skokan [19] extended this result to k-uniform hypergraphs. W.T. Gowers [4, 5] gave another such extension, using a different concept of regularity than that of Frankl, R¨odl and Skokan. Similarly to the graph case, all of these proofs are non-constructive. In this paper, we report on a constructive proof of Gowers’ Hypergraph Regularity Lemma, and discuss an application.

• Type: Journal
• Year: 2016
• Where: SODA
• Authors: Brendan Nagle, Vojtech Rödl, Mathias Schacht
• Keywords: Algorithms, SODA 2016