Entropy and expansion

Csóka, Endre; Harangi, Viktor; Virág, Bálint

doi:10.1214/19-aihp1044

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…This clearly produces an independent set, and it is not difficult to prove that each vertex v is selected with probability at least 1 d(v)+1 [4], so in particular this 1-round randomized algorithm witnesses the fact that the fractional chromatic number of graphs of maximum degree ∆ is at most ∆ + 1. Note that factor-of-IID algorithms for independent sets introduced in the past years are of this form (see for instance [9,10]). This leaves the question of how to produce a deterministic distributed algorithm for fractional coloring.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, it was proved that d-dimensional grids can be colored with 4 colors in O(log * n) rounds, while computing a 3-coloring in a 2-dimensional n × n-grid takes Ω(n) rounds [6]. For (almost) vertex-transitive graphs like grids finding minimum fractional colorings is essentially equivalent to finding maximum independent sets, and simple local randomized algorithms approaching the optimal independent set in grids can be used to produce (2 + ε)-fractional colorings with small output (see for instance [10]). In Section 4, we will show that for any fixed ε > 0 and…”

Section: Introductionmentioning

confidence: 99%

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Distributed algorithms for fractional coloring

Bousquet¹,

Pirot²

2020

Preprint

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In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It is known that for every real α > 1 and integer ∆, a fractional coloring of total weight at most α(∆ + 1) can be obtained deterministically in a single round in graphs of maximum degree ∆, in the LOCAL model of computation. However, a major issue of this result is that the output of each vertex has unbounded size. Here we prove that even if we impose the more realistic assumption that the output of each vertex has constant size, we can find fractional colourings with a weight arbitrarily close to known tight bounds for the fractional chromatic number in several cases of interest. Moreover, we improve on classical bounds on the chromatic number by considering the fractional chromatic number instead, without significantly increasing the output size and the round complexity of the existing algorithms.N. Bousquet, L. Esperet and F. Pirot are supported by ANR Projects GATO (ANR-16-CE40-0009-01) and GrR (ANR-18-CE40-0032).

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Distributed algorithms for fractional coloring

Bousquet¹,

Pirot²

2020

Preprint

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“…Several necessary conditions have been formulated for typical processes in the last years. Some of them are about the covariance structure, others are entropy inequalities [4,5,15]. However, general sufficient conditions for a process to be typical are less common.…”

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confidence: 99%

Typicality and entropy of processes on infinite trees

Bordenave

Backhausz

Szegedy

2022

Ann. Inst. H. Poincaré Probab. Statist.

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Consider a uniformly sampled random d-regular graph on n vertices. If d is fixed and n goes to ∞ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called "typical" processes) on the infinite d-regular tree T d . This correspondence between ergodic theory on T d and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on T d . It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees. Abstract.[Fr] On considère un graphe d-régulier aléatoire avec n sommets uniformément distribué. Si d est fixé et n diverge, nous pouvous alors relié les propriétés typiques (de grande probabilité) d'un tel graphe aléatoire avec une famille de processus aléatoires invariants (dénommés processus "typiques") sur l'arbre d-régulier infini T d . Cette correspondance entre théorie ergodique sur T d et graphes réguliers aléatoires s'est déjà révélée fructueuse dans les deux directions. Ce papier poursuit l'investigation des processus typiques avec un accent mis sur l'entropie. Nous y étudions une notion naturelle d'entropie micro-état pour les processus invariant sur T d . Elle sert de rafinement quantitatif à la notion de typicalité et elle est intimement reliée à l'energie libre asymptotique en physique statistique. Au moyen d'inégalités entropiques, nous démontrons des nouvelles conditions suffisantes de typicalité pour des processus markovien sur les arêtes de l'arbre. Nous étendons aussi ces notions et résultats à des processus sur des arbres de Galton-Watson unimodulaires.

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Distributed Algorithms for Fractional Coloring

Bousquet

Pirot

2021

Structural Information and Communication Complexity

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Entropy and expansion

Cited by 6 publications

References 24 publications

Distributed algorithms for fractional coloring

Distributed algorithms for fractional coloring

Typicality and entropy of processes on infinite trees

Distributed Algorithms for Fractional Coloring

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