Invariant Gaussian processes and independent sets on regular graphs of large girth

Csóka, Endre; Gerencsér, Balázs; Harangi, Viktor; Virág, Bálint

doi:10.1002/rsa.20547

Cited by 50 publications

(99 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is worth noting that Lyons’ result also applys to maximum bisection for which to our best knowledge his result is still the state‐of‐the‐art. Our improved bound is based on a simple argument taking advantage of a recent result by Csóka et al regarding the size of a largest bi‐partite subgraph of a cubic graph with large girth. We obtainTheorem Let

{double-struckG}_{n}

be an arbitrary sequence of n‐node cubic connected graphs with girth diverging to infinity.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Lemma 1.2 For every x satisfying (5) and β ∈ (0, 1/2), the system (7), (8), and (9) has a unique solution.…”

Section: Our Contributionmentioning

confidence: 99%

“…This lemma follows from Lemma 5.1 whose proof is given in Section 5. Given x and β ∈ (0, 1/2), denote the unique solution to (7), (8), and (9) by θ * 1 (x, β), θ * 2 (x, β), and t * (x, β). We introduce the following functions…”

Section: Our Contributionmentioning

confidence: 99%

“…Finally, we numerically compute x l in (13) based on the bisection method, in which for a given x satisfying (5) we use the command "FindRoot" in Mathematica to search for a solution to the equation system (7)(8)(9) and L(x, β, t, θ 1 , θ 2 ) = 2w(x) inside the region β ∈ [10 −10 , 1/4 − 10 −10 ]. If the search succeeds, we set x as an upper bound of x l , otherwise we set x as a lower bound of x l .…”

Section: Solving the Optimization Problem (112)mentioning

confidence: 99%

See 3 more Smart Citations

On the max‐cut of sparse random graphs

Gamarnik

2017

Random Struct Algorithms

View full text Add to dashboard Cite

We consider the problem of estimating the size of a maximum cut (Max-Cut problem) in a random Erdős-Rényi graph on n nodes and ⌊cn⌋ edges. It is shown in Coppersmith et al. [CGHS04] that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region [c/2 + 0.37613 √ c, c/2 + 0.58870 √ c] with high probability (w.h.p.) as n increases, for all sufficiently large c. The upper bound was obtained by application of the first moment method, and the lower bound was obtained by constructing algorithmically a cut which achieves the stated lower bound.In this paper we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size of the maximum cut normalized by the number of nodes belongs to the interval [c/2 + 0.47523 √ c, c/2 + 0.55909 √ c] w.h.p. as n increases, for all sufficiently large c. Instead of considering the expected number of cuts achieving a particular value as is done in the application of the first moment method, we observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved two dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value c/2+0.47523 √ c. It is worth noting that both bounds are stronger than the ones obtained by standard first and second moment methods.Finally, we also obtain an improved lower bound of 1.36000n on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of 1.33773n.

show abstract

{double-struckG}_{n}

be an arbitrary sequence of n‐node cubic connected graphs with girth diverging to infinity.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Lemma 1.2 For every x satisfying (5) and β ∈ (0, 1/2), the system (7), (8), and (9) has a unique solution.…”

Section: Our Contributionmentioning

confidence: 99%

Section: Our Contributionmentioning

confidence: 99%

Section: Solving the Optimization Problem (112)mentioning

confidence: 99%

See 2 more Smart Citations

On the max‐cut of sparse random graphs

Gamarnik

2017

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

“…This was improved by Lyons [20], who proved a lower bound of 0.89m for cubic graphs of girth at least 655. The best known lower bound for cubic graphs of large girth, 0.90m, was proved by Gamarnik and Li [10], using a result of Csóka, Gerencsér, Harangi, and Virág [5]. The bound of Lyons [20] holds for any d-regular graphs of large enough (but constant) girth: such graphs have a cut of size at least m · (…”

Section: Introductionmentioning

confidence: 99%

Local approximation of the Maximum Cut in regular graphs

Bamas

2020

Theoretical Computer Science

View full text Add to dashboard Cite

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.

show abstract

On large‐girth regular graphs and random processes on trees

Backhausz

Szegedy

2018

Random Struct Algorithms

View full text Add to dashboard Cite

We study various classes of random processes defined on the regular tree T d that are invariant under the automorphism group of T d . The most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. K E Y W O R D Sfactor of i.i.d.

show abstract

Invariant Gaussian processes and independent sets on regular graphs of large girth

Cited by 50 publications

References 10 publications

On the max‐cut of sparse random graphs

On the max‐cut of sparse random graphs

Local approximation of the Maximum Cut in regular graphs

On large‐girth regular graphs and random processes on trees

Contact Info

Product

Resources

About