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## Approximate Hypergraph Partitioning and Applications

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Citations: | 7 - 1 self |

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475 | Property testing and its connection to learning and approximation.
- Goldreich, Goldwasser, et al.
- 1998
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Citation Context ... Theorem 1 and Theorem 2 Before going to the formal proof of Theorem 1 and Theorem 2 we briefly describe the general idea of how it is done. The outermost layer of the proof borrows from the proof of =-=[16]-=- for graph partitions. Assuming that the hypergraph admits a partition Π = {V1, . . . , Vk} of the vertices with the desired densities, we first split V into ℓ = O(1/ɛ) parts Y 1 , Y 2 , . . . , Y ℓ o... |

257 | Szemeredi’s regularity lemma and its applications in graph theory, Combinatorics, Paul Erdos is eighty,
- Komlos, Simonovits
- 1993
(Show Context)
Citation Context ...applications since dealing with random-like graphs is much easier than dealing with arbitrary graphs. For a comprehensive survey of the applications of the lemma, the interested reader is referred to =-=[22]-=-. We first state the lemma. Recall that for two nonempty disjoint vertex sets A and B of a graph G, we define E(A, B) to be the set of edges of G between A and B. The edge density of the pair is defin... |

212 | The Probabilistic Method. Second Edition - Alon, Spencer - 2000 |

189 | Polynomial time approximation schemes for dense instances of NP-hard problems.
- Arora, Karger, et al.
- 1995
(Show Context)
Citation Context ...r-science. Standard examples of partition problems include k-colorability, Max-Clique and Max-Cut. Most of these problems are computationally hard even to approximate, but it was observed in the 90’s =-=[5, 11]-=- that many of these partition problems have good approximations when the input graph is dense. In this paper we introduce an efficient O(n) algorithm for partitioning hypergraphs, with an accompanying... |

151 | Hypergraph regularity and the multidimensional Szemerédi theorem - Gowers - 2007 |

148 |
Regular partitions of graphs, in:
- Szemerédi
- 1976
(Show Context)
Citation Context ... problems making only poly(1/ɛ) queries, and with a constant running time. We present several applications of our result, and in the foremost a new application related to Szemerédi’s regularity lemma =-=[27]-=-. By using an appropriate hypergraph modeling of the problem we design a surprisingly simple O(n) time algorithm for constructing regular partitions of graphs. An added benefit is that unlike the prev... |

139 | Approximating the cut-norm via Grothendieck’s inequality.
- Alon, Naor
- 2006
(Show Context)
Citation Context ...tioning algorithm to obtain the following results: • We derive a surprisingly simple O(n) time algorithmic version of Szemerédi’s regularity lemma. Unlike all the previous approaches for this problem =-=[3, 10, 14, 15, 21]-=-, which reproved the lemma “algorithmically”, and only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists. • For any r ≥ 3, we g... |

108 | The algorithmic aspects of the Regularity Lemma,
- Alon, Lefmann, et al.
- 1994
(Show Context)
Citation Context ...ply an efficient polynomial time algorithm for obtaining a regular partition of the graph. The first polynomial time algorithm for constructing such a partition of a graph was obtained by Alon et al. =-=[2]-=-. Additional algorithmic versions of the lemma were later obtained in [3, 10, 14, 15, 21]. 7sThe main observation used in most of the above was that although subset-regularity is computationally hard ... |

105 | The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms
- Nagle, Rödl, et al.
(Show Context)
Citation Context ...( 1 ɛ )2 m · � skr · log( ) δ � + m · T C(Ψ) 8 Concluding Remarks and Open Problems Strong hypergraph regularity: Very recently, a lot of attention was given to obtaining hypergraph regularity lemmas =-=[17, 24, 25, 28]-=- that are strong enough to reprove Szemerédi’s numbertheoretic theorem [26] and to prove combinatorial deletion results. As it turns out, in these lemmas one needs to consider not only partitions of t... |

90 | Regularity lemma for k-uniform hypergraphs.
- Rodl, Skokan
- 2004
(Show Context)
Citation Context ...( 1 ɛ )2 m · � skr · log( ) δ � + m · T C(Ψ) 8 Concluding Remarks and Open Problems Strong hypergraph regularity: Very recently, a lot of attention was given to obtaining hypergraph regularity lemmas =-=[17, 24, 25, 28]-=- that are strong enough to reprove Szemerédi’s numbertheoretic theorem [26] and to prove combinatorial deletion results. As it turns out, in these lemmas one needs to consider not only partitions of t... |

89 |
The Regularity Lemma and approximation schemes for dense problems
- Frieze, Kannan
- 1996
(Show Context)
Citation Context ...tioning algorithm to obtain the following results: • We derive a surprisingly simple O(n) time algorithmic version of Szemerédi’s regularity lemma. Unlike all the previous approaches for this problem =-=[3, 10, 14, 15, 21]-=-, which reproved the lemma “algorithmically”, and only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists. • For any r ≥ 3, we g... |

80 |
Lower bounds of tower type for Szemeredi’s uniformity lemma.
- Gowers
- 1997
(Show Context)
Citation Context ... an enormous dependence on ɛ. More precisely, denote the tower of exponents function as Tower(i) = 2 Tower(i−1) = 2 ···2�i . Then the bounds on T4.2(ɛ) are given by Tower(1/ɛ 5 ). Furthermore, Gowers =-=[17]-=- proved that these bounds are not far from the truth (in the qualitative sense), as he constructed a graph that has no ɛ-regular partition of size less than Tower(1/ɛ 1/16 ). Of course, this takes a h... |

75 | A variant of the hypergraph removal lemma
- Tao
(Show Context)
Citation Context ...( 1 ɛ )2 m · � skr · log( ) δ � + m · T C(Ψ) 8 Concluding Remarks and Open Problems Strong hypergraph regularity: Very recently, a lot of attention was given to obtaining hypergraph regularity lemmas =-=[17, 24, 25, 28]-=- that are strong enough to reprove Szemerédi’s numbertheoretic theorem [26] and to prove combinatorial deletion results. As it turns out, in these lemmas one needs to consider not only partitions of t... |

61 |
Quick approximation to matrices and applications, Combinatorica 19
- Frieze, Kannan
- 1999
(Show Context)
Citation Context ...rior work on hypergraph partition algorithms. Algorithms similar to the hypergraph partition algorithm that we obtain here have been considered in some earlier papers. Most notably, Frieze and Kannan =-=[13]-=- and Andersson and En4 To be precise, the exact bounds we quote for Max-Cut and 3-colorability follow from a more specialized argument given in [16] and not directly from the general GGR algorithm. Se... |

58 | Graph Theory, Third Edition. - Diestel - 2006 |

54 |
Regularity lemmas for hypergraphs and quasi-randomness.
- Chung
- 1991
(Show Context)
Citation Context ...larity counter examples for the irregular r-tuples, and works almost word for word for hypergraphs. We will not reproduce it in this version as it has been proved in several previous papers, see e.g. =-=[6]-=- and [8]. Lemma 5.6 For every r and ɛ there exist δ = δ5.6(r, ɛ) and f(k) = f (r,ɛ) 5.6 (k) so that any equipartition of an r-uniform hypergraph that is (δ, f)-robust is also ɛ-regular. The above lemm... |

42 |
A fast approximation algorithm for computing the frequencies of subgraphs in a given graph
- Duke, Lefmann, et al.
- 1995
(Show Context)
Citation Context ...tioning algorithm to obtain the following results: • We derive a surprisingly simple O(n) time algorithmic version of Szemerédi’s regularity lemma. Unlike all the previous approaches for this problem =-=[3, 10, 14, 15, 21]-=-, which reproved the lemma “algorithmically”, and only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists. • For any r ≥ 3, we g... |

38 | Hypergraphs, quasi-randomness and conditions for regularity - Kohayakawa, Rödl, et al. |

35 | Testing versus estimation of graph properties
- Fischer, Newman
(Show Context)
Citation Context ...ition, that of Definition 4.3. The framework of the graph partition problems of [16] thus cannot by itself provide a check for regularity (unless “negations” of partition properties are checked as in =-=[12]-=- and in the hypergraph regularity algorithms here, but these again may lead to a small regular partition being overlooked). However, as we show below, a hypergraph partition theorem such as Theorem 1 ... |

33 |
de la Vega, MAX-CUT has a Randomized Approximation Scheme
- Fernandez
- 1996
(Show Context)
Citation Context ...r-science. Standard examples of partition problems include k-colorability, Max-Clique and Max-Cut. Most of these problems are computationally hard even to approximate, but it was observed in the 90’s =-=[5, 11]-=- that many of these partition problems have good approximations when the input graph is dense. In this paper we introduce an efficient O(n) algorithm for partitioning hypergraphs, with an accompanying... |

26 | Random sampling and approximation of MAX-CSP problems.
- Alon, Vega, et al.
- 2002
(Show Context)
Citation Context ...partition problem can be easily used to derive some results that were previously proved using specialized methods, namely testing properties of hypergraphs [7, 23] and estimating k-CNF satisfiability =-=[1]-=-. 2 Extension of the GGR-algorithm Our main result in this paper is a generalization of the GGR-algorithm to the case of hypergraphs. Let us start by defining our framework for studying hypergraph par... |

21 | An optimal algorithm for checking regularity
- Kohayakawa, Rödl, et al.
- 2003
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Citation Context |

20 |
Integer sets containing no k elements in arithmetic progression
- Szemerédi
- 1975
(Show Context)
Citation Context ...stigate here is the one discussed e.g. in [15] (the “vertex partition” version), which is not strong enough for some applications such as proving Szemerédi’s Theorem on r-term arithmetic progressions =-=[26]-=-, but still has many applications. For example, it was used in [15] in order to obtain additive approximations for all Max-SNP problems. This regularity is defined in an analog manner to the definitio... |

19 | An algorithmic regularity lemma for hypergraphs.,
- Czygrinow, Rödl
- 2000
(Show Context)
Citation Context ...exists. • For any r ≥ 3, we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n 2r−1 ) time (deterministic) algorithms =-=[8, 15]-=-. The property testing algorithm is used to unify several previous results, and to obtain the partition densities for the above problems (rather than the partitions themselves) using only poly(1/ɛ) qu... |

18 | A simple algorithm for constructing Szemerédi’s regularity partition
- Frieze, Kannan
- 1999
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Citation Context |

11 | An algorithmic version of the hypergraph regularity lemma
- Haxell, Nagle, et al.
- 2008
(Show Context)
Citation Context ...consider various requirements on the interactions between the partitions of different arities concerning their densities and beyond (e.g. on the count of certain small substructures). See for example =-=[19]-=- for an algorithmic version of a strong regularity lemma for 3-uniform hypergraphs. 30sWe believe that we can extend our main result about vertex partitions to the case of partitions of pairs and beyo... |

5 |
Property testers for dense constraint satisfaction programs on finite domains, Random Structures Algorithms 21
- Andersson, Engebretsen
- 2002
(Show Context)
Citation Context ...ends the result of Goldreich, Goldwasser and Ron for graph partition problems [16], and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of =-=[5]-=-. ∗ A PRELIMINARY VERSION OF THIS PAPER APPEARED IN PROC. OF THE 48 T H ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS) 2007, 579589. †Faculty of Computer Science, Technion – Israel in... |

2 |
Testing hypergraph colorability, Theor
- Czumaj, Sohler
(Show Context)
Citation Context ... show how special cases of the now-testable partition problem can be easily used to derive some results that were previously proved using specialized methods, namely testing properties of hypergraphs =-=[7, 23]-=- and estimating k-CNF satisfiability [1]. 2 Extension of the GGR-algorithm Our main result in this paper is a generalization of the GGR-algorithm to the case of hypergraphs. Let us start by defining o... |

1 |
Testing the independence number of hypergraphs
- Langberg
(Show Context)
Citation Context ... show how special cases of the now-testable partition problem can be easily used to derive some results that were previously proved using specialized methods, namely testing properties of hypergraphs =-=[7, 23]-=- and estimating k-CNF satisfiability [1]. 2 Extension of the GGR-algorithm Our main result in this paper is a generalization of the GGR-algorithm to the case of hypergraphs. Let us start by defining o... |