Superfast coloring in CONGEST via efficient color sampling

Halldórsson, Magnús M.; Nolin, Alexandre

doi:10.1016/j.tcs.2023.113711

Cited by 1 publication

(3 citation statements)

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“…For vertex-coloring at distance two, when (1 + ε)∆ 2 colors are available, the problem is much easier, and is known to be solvable in O(log 4 log n) rounds [HN23]. The first poly(log n)-round CONGEST algorithm for distance-2 (∆ 2 + 1)-coloring was given in [HKM20], while a O(log ∆) + poly(log log n)-round algorithm was given in [HKMN20].…”

Section: Related Workmentioning

confidence: 99%

“…At the outset, several parts of this schema already exist for distance-2 coloring. In particular, generating slack is trivial, a poly(log log n)-round algorithm for coloring poly(log n)-degree graphs is known from [HKMN20], and coloring with slack Ω(∆ 2 ) follows from [HN23]. Almost-clique decompositions (ACD) have been well studied and need only a minor tweak here.…”

Section: Our Techniques In a Nutshellmentioning

confidence: 99%

“…Coloring Sparse Nodes (Steps 2 & 3). The coloring of sparse nodes was already handled in [HN23]. After GenerateSlack, all sparse nodes have slack proportional to ∆ 2 (Proposition 2.3).…”

Section: Pseudo-degreesmentioning

confidence: 99%

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We provide a O(log 6 log n)-round randomized algorithm for distance-2 coloring in CONGEST with ∆ 2 +1 colors. For ∆ ≫ poly log n, this improves exponentially on the O(log ∆+poly log log n) algorithm of [Halldórsson, Kuhn, Maus, Nolin, DISC'20].Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lovász Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.

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Section: Related Workmentioning

confidence: 99%

Section: Our Techniques In a Nutshellmentioning

confidence: 99%