Maximum edge‐cuts in cubic graphs with large girth and in random cubic graphs

Kardoš, František; Kráľ, Daniel; Volec, Jan

doi:10.1002/rsa.20471

Cited by 14 publications

(23 citation statements)

References 24 publications

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“…Also evolving from the early version of our method in [23], Kardoš, Král and Volec [30] found strong lower bounds on the size of maximum cuts in cubic graphs with large girth, which improved on earlier bounds of Zýka [51], and the first author [24] found lower bounds on the largest induced forest in regular graphs with large girth.…”

Section: Related Work and Examples Of Improved Boundsmentioning

confidence: 71%

“…Strangely perhaps, relaxing the independence condition does not produce any easy significant improvement. For maximum cuts in cubic graphs, we derive analytic lower bounds, based on differential equations, such that the bounds obtained in [30] can be interpreted as long recurrences for computing numerical approximations of our analytic result.…”

Section: Related Work and Examples Of Improved Boundsmentioning

confidence: 99%

“…This used a computation that numerically iterated a recurrence g 0 times. Since the method was based on [23], which employed some of the ideas in the present paper (including some use of deprioritised algorithms), it is not too surprising that the algorithm in [30] is a chunky local deletion algorithm that derives in a natural way from the prioritised algorithm A described at the end of Section 4.1. (However, A is not directly described in [30].)…”

Section: Large Cuts In Random Cubic Graphsmentioning

confidence: 99%

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Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

Hoppen

Wormald

2018

Combinatorica

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We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This is an uncommon direction of transfer of results, which is usually from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As examples, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum and minimum bisection, maximum k-independent sets, minimum k-dominating sets and minimum connected and weakly-connected dominating sets in r-regular graphs with large girth. * Supported by FAPERGS (Proc. 11/1436-1) and CNPq (Proc. 486108/2012-0 and 304510/2012-2).

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Section: Related Work and Examples Of Improved Boundsmentioning

confidence: 71%

Section: Related Work and Examples Of Improved Boundsmentioning

confidence: 99%

Section: Large Cuts In Random Cubic Graphsmentioning

confidence: 99%

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Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

Hoppen

Wormald

2018

Combinatorica

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“…The case where d is small and the girth (length of a shortest cycle) is large has also been considered: for 3-regular graphs, Kardoš, Král' and Volec [17] showed that when the girth is at least 637789, there exists a randomized distributed algorithm that outputs a cut of average size at least 0.88672m in at most 318894 rounds (the important value here is the size of the cut). This was improved by Lyons [20], who proved a lower bound of 0.89m for cubic graphs of girth at least 655.…”

Section: Introductionmentioning

confidence: 99%

Local approximation of the Maximum Cut in regular graphs

Bamas

2020

Theoretical Computer Science

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This paper is devoted to the distributed complexity of finding an approximation of the maximum cut (MaxCut) in graphs. A classical algorithm consists in letting each vertex choose its side of the cut uniformly at random. This does not require any communication and achieves an approximation ratio of at least 1 2 in expectation. When the graph is d-regular and triangle-free, a slightly better approximation ratio can be achieved with a randomized algorithm running in a single round. Here, we investigate the round complexity of deterministic distributed algorithms for MaxCut in regular graphs. We first prove that if G is d-regular, with d even and fixed, no deterministic algorithm running in a constant number of rounds can achieve a constant approximation ratio. We then give a simple one-round deterministic algorithm achieving an approximation ratio of 1 d for d-regular graphs when d is odd. We show that this is best possible in several ways, and in particular no deterministic algorithm with approximation ratio 1 d + (with > 0) can run in a constant number of rounds. We also prove results of a similar flavour for the MaxDiCut problem in regular oriented graphs, where we want to maximize the number of arcs oriented from the left part to the right part of the cut.

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“…Indeed, this is the expected size of a uniformly random cut. Moreover, though there are certain classes of graphs for which the max-cut is significantly larger (see, for example, [22]), the max-cut of an m-edge graph is typically quite close to m/2. As such, much of the focus has been on maximising the excess of a cut, defined to be its size minus m/2.…”

Section: Introductionmentioning

confidence: 99%

Hypergraph cuts above the average

et al. 2019

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An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2 + Ω( √ m) and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is Ω( √ m) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every medge k-uniform hypergraph has an r-cut whose size is Ω(m 5/9 ) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

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Maximum edge‐cuts in cubic graphs with large girth and in random cubic graphs

Cited by 14 publications

References 24 publications

Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

Local approximation of the Maximum Cut in regular graphs

Hypergraph cuts above the average

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