Distributed Algorithms for Fractional Coloring

Bousquet, Nicolás; Pirot, François

doi:10.1007/978-3-030-79527-6_2

Cited by 2 publications

(11 citation statements)

References 21 publications

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“…Similar results were also shown in [24]. Very recently, Bousquet, Esperet, and Pirot [11] made some interesting further progress on the distributed fractional coloring problem.…”

Section: Introduction and Related Worksupporting

confidence: 83%

“…Our first main contribution, presented in Section 5, is in the improvement of the main algorithm of [11]. In particular, we start by showing that a fractional (∆ + ε)-coloring, with small support, can be obtained in a time that depends only polynomially in ∆ and ε.…”

Section: Our Resultsmentioning

confidence: 99%

“…In [11], it is shown that although a fractional (∆ + 1)-coloring can be computed in a single communication round, in ∆-regular graphs that do not contain K ∆+1 as a subgraph, the fractional ∆-coloring problem requires time Ω(log ∆ n) deterministically and Ω(log ∆ log n) rounds with randomization. That is, for the fractional ∆-coloring problem, the same lower bounds as for the standard ∆-coloring problem hold.…”

Section: Introduction and Related Workmentioning

confidence: 99%

“…In this paper, we improve on the results of [11] in several ways. We here give an overview over our results, for a detailed statement of the results, we refer to Section 3.…”

Section: Introduction and Related Workmentioning

confidence: 99%

“…First, we improve the time to compute a fractional (∆ + ε)-coloring. We show that a coloring of the same quality as in [11] (i.e., a (q∆ + 1 : q)-coloring for q = 1/ε) can be computed in time…”

Section: Introduction and Related Workmentioning

confidence: 99%

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Improved Distributed Fractional Coloring Algorithms

Balliu¹,

Kühn²,

Olivetti³

2021

Preprint

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We prove new bounds on the distributed fractional coloring problem in the LOCAL model. A fractional c-coloring of a graph G = (V, E) is a fractional covering of the nodes of G with independent sets such that each independent set I of G is assigned a fractional value λI ∈ [0, 1]. The total value of all independent sets of G is at most c, and for each node v ∈ V , the total value of all independent sets containing v is at least 1. Equivalently, fractional c-colorings can also be understood as multicolorings as follows. For some natural numbers p and q such that p/q ≤ c, each node v is assigned a set of at least q colors from {1, . . . , p} such that adjacent nodes are assigned disjoint sets of colors. The minimum c for which a fractional c-coloring of a graph G exists is called the fractional chromatic number χ f (G) of G.Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant ε > 0, a fractional (∆ + ε)-coloring can be computed in ∆ O(∆) + O(∆ • log * n) rounds. We show that such a coloring can be computed in only O(log 2 ∆) rounds, without any dependency on n.We further show that in O log n ε rounds, it is possible to compute a fractional (1 + ε)χ f (G)coloring, even if the fractional chromatic number χ f (G) is not known. That is, the fractional coloring problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model. For the standard coloring problem, it is only known that an O log n log log n -approximation can be computed in polylogarithmic time in the LOCAL model. We also show that our distributed fractional coloring approximation algorithm is best possible. We show that in trees, which have fractional chromatic number 2, computing a fractional (2 + ε)-coloring requires at least Ω log n ε rounds. We finally study fractional colorings of regular grids. In [Bousquet, Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded dimension, a fractional (2 + ε)-coloring can be computed in time O(log * n). We show that such a coloring can even be computed in O(1) rounds in the LOCAL model.

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“…Similar results were also shown in [24]. Very recently, Bousquet, Esperet, and Pirot [11] made some interesting further progress on the distributed fractional coloring problem.…”

Section: Introduction and Related Worksupporting

confidence: 83%

Section: Our Resultsmentioning

confidence: 99%

Section: Introduction and Related Workmentioning

confidence: 99%

“…In this paper, we improve on the results of [11] in several ways. We here give an overview over our results, for a detailed statement of the results, we refer to Section 3.…”

Section: Introduction and Related Workmentioning

confidence: 99%

Section: Introduction and Related Workmentioning

confidence: 99%

See 3 more Smart Citations