On Regularity Lemmas and their Algorithmic Applications

Abstract

Abstract. Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze-Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze-Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ǫ ′ > ǫ, an ǫ ′ -regular partition with k parts if there exists an ǫ-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

Fast Property Testing and Metrics for Permutations

“…By conditions (i) and (ii), we have for 0 ≤ i ≤ K, n i,i = n i and n i,t = 0 for t < j. For i ≥ K + 1, n i,t = 0 for t < K. Thus combining with Lemma 4.2(c) for the bound on n i , the first summand in (14) can be bounded by…”

“…This regularity lemma was subsequently improved upon by Hoppen, Kohayakawa, and Sampaio[19] and with much better quantitative estimates by Fox, Lovász, and Zhao[14].…”

“…The rectangular distance for permutations is an analogue of the Frieze-Kannan cut distance for graphs, which has played an important role in the development of the weak regularity lemma, graph limits, and approximation algorithms for graphs. See, for example, the book by Lovász [25] and the paper [14].…”

Regular decomposition of large graphs and other structures: Scalability and robustness towards missing data

“…In future work we try to find proofs and conditions for the described sampling scheme assuming also partially lost data and without the assumption that graph is generated using a SBM. Here the 'Sampling lemma' 2.3 from [4] can be handy, proving that in general setting the link densities can be found from small samples. Even more intriguing is the question of using RD for sparse graphs and finding some kind of analog of RD-approach in this situation.…”

“…However, a similar result is extended also to many other cases and structures, see the Refs. in [4] with a constant flow of significant new results.…”