A Deterministic Algorithm for the Frieze–Kannan Regularity Lemma

Dellamonica, Domingos; Kalyanasundaram, Subrahmanyam; Martin, Daniel M.; Rödl, Vojtěch; Shapira, A.

doi:10.1137/110846373

Cited by 9 publications

(24 citation statements)

References 22 publications

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“…For example, in the algorithmic version [8], the first step (also the key step) is given as the following result [8, Corollary 3.1].…”

Section: Algorithmic Weak Regularitymentioning

confidence: 99%

“…In Theorems 1.1 and 1.2 and Corollary 1.3, the dependence of the running time on the parameters ε, α, k may be improved at the cost of worsening the dependence on n from n 2 to n ω+o (1) . This is because we use the recent algorithmic version of the Frieze-Kannan weak regularity lemma due to Dellamonica, Kalyanasundaram, Martin, Rödl and Shapira [8,9]. In the more recent paper [9], they develop an O ε (n 2 ) algorithm for finding a weak ε-regular partition, but it has a double exponential in 1/ε constant factor dependence.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Regularity Lemmas and their Algorithmic Applications

Fox¹,

Lovász²,

Zhao³

2017

Combinator. Probab. Comp.

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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.

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“…For example, in the algorithmic version [8], the first step (also the key step) is given as the following result [8, Corollary 3.1].…”

Section: Algorithmic Weak Regularitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Regularity Lemmas and their Algorithmic Applications

Fox¹,

Lovász²,

Zhao³

2017

Combinator. Probab. Comp.

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show abstract

“…which takes into account the edge weights. More recent algorithms for finding regular partitions can be found in [15], [22], [23]. The algorithmic solutions developed so far have been focused exclusively on exact algorithms whose worst-case complexity, although being polynomial in the size of the underlying graph, has a hidden tower-type dependence on an accuracy parameter.…”

Section: Finding Regular Partitionsmentioning

confidence: 99%

Revealing structure in large graphs: Szemerédi’s regularity lemma and its use in pattern recognition

Pelillo

Elezi

Fiorucci

2017

Pattern Recognition Letters

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Introduced in the mid-1970's as an intermediate step in proving a long-standing conjecture on arithmetic progressions, Szemerédi's regularity lemma has emerged over time as a fundamental tool in different branches of graph theory, combinatorics and theoretical computer science. Roughly, it states that every graph can be approximated by the union of a small number of random-like bipartite graphs called regular pairs. In other words, the result provides us a way to obtain a good description of a large graph using a small amount of data, and can be regarded as a manifestation of the all-pervading dichotomy between structure and randomness. In this paper we will provide an overview of the regularity lemma and its algorithmic aspects, and will discuss its relevance in the context of pattern recognition research.

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“…either (1) correctly states that d (G, G ) ε, or (2) outputs sets S and T such that |e G (S, T ) − e G (S, T )| > ε −O (1) . In particular, the fact that G may not have bounded weights presents a challenge in applying results from [1,2].…”

mentioning

confidence: 99%

“…Theorem 1. There is a deterministic algorithm that, given ε > 0 and an n-vertex graph G, outputs, in ε −O (1)…”

mentioning

confidence: 99%

Erratum: On Regularity Lemmas and their Algorithmic Applications

Fox¹,

Lovász²,

Zhao³

2018

Combinator. Probab. Comp.

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We retract Corollary 3.5. We claimed that it follows from a recent algorithmic version [1, 2] of the Frieze-Kannan regularity lemma (see Theorems 3.2 and Theorem 3.3 in [3]). Unfortunately there is an error in our application of these algorithmic results. Although the algorithms in [1, 2] can test whether a partition is weakly regular, it is unclear how to apply them to an intermediate step in our proposed algorithm. Specifically, we would need an algorithm that, given

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A Deterministic Algorithm for the Frieze–Kannan Regularity Lemma

Cited by 9 publications

References 22 publications

On Regularity Lemmas and their Algorithmic Applications

On Regularity Lemmas and their Algorithmic Applications

Revealing structure in large graphs: Szemerédi’s regularity lemma and its use in pattern recognition

Erratum: On Regularity Lemmas and their Algorithmic Applications

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