An Optimal Algorithm for Finding Frieze–Kannan Regular Partitions

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

doi:10.1017/s0963548314000200

Cited by 10 publications

(23 citation statements)

References 14 publications

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“…Recall that ω < 2.373 is the matrix multiplication exponent. In [9] they gave a different algorithm which improved the dependence of the running time on n from O ε (n ω+o (1) ) to O ε (n 2 ), while sacrificing the dependence of ε. Namely, it was shown that there is a deterministic algorithm that finds, in…”

Section: Algorithmic Weak Regularitymentioning

confidence: 99%

“…In [9], an alternative theorem is given for finding an irregular pair. The statement below is a consequence of [9, Theorem 8.1].…”

Section: Theorem 32mentioning

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 2 more Smart Citations

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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Section: Algorithmic Weak Regularitymentioning

confidence: 99%

“…In [9], an alternative theorem is given for finding an irregular pair. The statement below is a consequence of [9, Theorem 8.1].…”

Section: Theorem 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Regularity Lemmas and their Algorithmic Applications

Fox¹,

Lovász²,

Zhao³

2017

Combinator. Probab. Comp.

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“…We apply Theorem 4 to G with approximation parameter ε, and obtain an equitable partition P : V = V 1 ∪ · · · ∪ V K on our vertex set with the desired properties such that 1/ε < K < 2 ε −c = 2 √ log log n . This can be done deterministically in n 2+o(1) -time using the algorithm of Dellamonica et al [15]. 2.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Approximating the rectilinear crossing number

Fox¹,

Pach²,

Suk³

2019

Computational Geometry

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A straight-line drawing of a graph G is a mapping which assigns to each vertex a point in the plane and to each edge a straightline segment connecting the corresponding two points. The rectilinear crossing number of a graph G, cr(G), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating cr(G) appears to be a difficult problem, and deciding if cr(G) ≤ k is known to be NP-hard. In fact, the asymptotic behavior of cr(Kn) is still unknown.In this paper, we present a deterministic n 2+o(1) -time algorithm that finds a straight-line drawing of any n-vertex graph G with cr(G) + o(n 4 ) pairs of crossing edges. Together with the well-known Crossing Lemma due to Ajtai et al. and Leighton, this result implies that for any dense n-vertex graph G, one can efficiently find a straight-line drawing of G with (1 + o(1))cr(G) pairs of crossing edges.

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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%

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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An Optimal Algorithm for Finding Frieze–Kannan Regular Partitions

Cited by 10 publications

References 14 publications

On Regularity Lemmas and their Algorithmic Applications

On Regularity Lemmas and their Algorithmic Applications

Approximating the rectilinear crossing number

Erratum: On Regularity Lemmas and their Algorithmic Applications

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