Complete deterministic directed finite automaton with transition graph Γ
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Transcript of Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf ·...
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Deterministic Distributed Edge-Coloring via
Hypergraph Maximal Matching
Manuela FischerETH Zurich
Mohsen GhaffariETH Zurich
Fabian KuhnUni Freiburg
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LOCAL Model (Linial [FOCS’87])
• undirected graph 𝐺 = 𝑉, 𝐸 , n nodes, maximum degree Δ
• synchronous message-passing rounds
• unbounded message size and computation
• Round Complexity: number of rounds to solve the problem
round complexity of a problem characterizes its locality
![Page 3: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/3.jpg)
LOCAL Model (Linial [FOCS’87])
• undirected graph 𝐺 = 𝑉, 𝐸 , n nodes, maximum degree Δ
• synchronous message-passing rounds
• unbounded message size and computation
• Round Complexity: number of rounds to solve the problem
round complexity of a problem characterizes its locality
![Page 4: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/4.jpg)
Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
![Page 5: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/5.jpg)
Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏 Linial [SICOMP’92]
![Page 6: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/6.jpg)
Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
𝛀 𝐥𝐨𝐠 𝒏/𝐥𝐨𝐠𝐥𝐨𝐠 𝒏
𝛀 𝐥𝐨𝐠 𝒏/𝐥𝐨𝐠𝐥𝐨𝐠 𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏 Linial [SICOMP’92]
𝛀 𝐥𝐨𝐠 𝒏/𝐥𝐨𝐠𝐥𝐨𝐠 𝒏 Kuhn, Moscibroda, Wattenhofer [PODC’06]
![Page 7: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/7.jpg)
Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
𝛀 𝐥𝐨𝐠 𝒏/𝐥𝐨𝐠𝐥𝐨𝐠 𝒏
𝛀 𝐥𝐨𝐠 𝒏/𝐥𝐨𝐠𝐥𝐨𝐠 𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏
𝛀 𝐥𝐨𝐠∗𝒏 Linial [SICOMP’92]
𝛀 𝐥𝐨𝐠 𝒏/𝐥𝐨𝐠𝐥𝐨𝐠 𝒏 Kuhn, Moscibroda, Wattenhofer [PODC’06]
𝐫𝐚𝐧𝐝𝐨𝐦𝐢𝐳𝐞𝐝 𝑶(𝐥𝐨𝐠 𝒏) Luby [STOC’85], Alon, Babai, Itai [JALG’86]
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Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
2O log n Panconesi, Srinivasan [STOC’92] 2O log n Panconesi, Srinivasan [STOC’92]
2O log n Panconesi, Srinivasan [STOC’92] 2O log n Panconesi, Srinivasan [STOC’92]
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Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
2O log n Panconesi, Srinivasan [STOC’92] 2O log n Panconesi, Srinivasan [STOC’92]
2O log n Panconesi, Srinivasan [STOC’92] 2O log n Panconesi, Srinivasan [STOC’92]
Linial’s Question from the 1980s:Is there an efficient (poly log n rounds) deterministic algorithm for Maximal Independent Set?
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Maximal Independent Set
Classic LOCAL Graph Problems
Maximal Matching
(Δ + 1)-Vertex-Coloring
(2Δ − 1)-Edge-Coloring
2O log n Panconesi, Srinivasan [STOC’92] 2O log n Panconesi, Srinivasan [STOC’92]
2O log n Panconesi, Srinivasan [STOC’92] 2O log n Panconesi, Srinivasan [STOC’92]
𝑂 log7 𝑛 Hańćkowiak, Karoński, Panconesi [SODA’98]
𝑂 log4 𝑛 Hańćkowiak, Karoński, Panconesi [PODC’99]
𝑂 log3 𝑛 F. [DISC’17]
Linial’s Question from the 1980s:Is there an efficient (poly log n rounds) deterministic algorithm for Maximal Independent Set?
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Linial’s Question from the 1980s:Is there an efficient (poly log n rounds) deterministic algorithm for Maximal Independent Set?
Classic LOCAL Graph Problems
"While maximal matchings can becomputed in polylogarithmic time […],
it is a decade old open problem whetherthe same running time is achievable for the
remaining three structures."
Panconesi, Rizzi ‘01
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Open Problem 11.4: Devise or rule out a deterministic(2𝛥 − 1)-edge-coloring algorithmthat runs in polylogarithmic time.
Barenboim, Elkin ‘13
Linial’s Question from the 1980s:Is there an efficient (poly log n rounds) deterministic algorithm for Maximal Independent Set?
Classic LOCAL Graph Problems
"While maximal matchings can becomputed in polylogarithmic time […],
it is a decade old open problem whetherthe same running time is achievable for the
remaining three structures."
Panconesi, Rizzi ‘01
![Page 13: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/13.jpg)
Overview of Results & Implications
(𝟐𝜟 − 𝟏)-Edge-Coloring in O log8 n rounds
Randomized (𝟐𝜟 − 𝟏)-Edge-Coloringin O log8 log n rounds
Open Problem 11.4
![Page 14: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/14.jpg)
Overview of Results & Implications
(𝟐𝜟 − 𝟏)-Edge-Coloring in O log8 n rounds
Randomized (𝟐𝜟 − 𝟏)-Edge-Coloringin O log8 log n rounds
𝟏 + 𝛆 -Approximation of Matching
Maximal Independent Set and 𝚫 + 𝟏 -Vertex-Coloringfor graphs with bounded neighborhood independence
Low-Out-Degree Orientation
Open Problem 11.5
almost Open Problem 11.10
Open Problem 11.4
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Overview of Results & Implications
Rank-r-Hypergraph Maximal Matching
in poly r ⋅ logO log r Δ ⋅ log n rounds
(𝟐𝜟 − 𝟏)-Edge-Coloring in O log8 n rounds
Randomized (𝟐𝜟 − 𝟏)-Edge-Coloringin O log8 log n rounds
𝟏 + 𝛆 -Approximation of Matching
Maximal Independent Set and 𝚫 + 𝟏 -Vertex-Coloringfor graphs with bounded neighborhood independence
Low-Out-Degree Orientation
Open Problem 11.5
almost Open Problem 11.10
Open Problem 11.4
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I) Formulation as Hypergraph Maximal Matching
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I) Formulation as Hypergraph Maximal Matching
II) Hypergraph Maximal Matching Algorithm
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I) Formulation as Hypergraph Maximal Matching
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(2Δ − 1)-Edge-Coloring
Maximal Independent Set
Unified Formulation as Hypergraph Maximal Matching Problem
Maximal Matching
(Δ + 1)-Vertex-Coloring
cast classic LOCAL graph problems as hypergraph maximal matching problems (LOCAL reductions)
rank 2 rank 3
rank Δ + 1rank Δ
![Page 20: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/20.jpg)
(2Δ − 1)-Edge-Coloring
Maximal Independent Set
Unified Formulation as Hypergraph Maximal Matching Problem
Maximal Matching
(Δ + 1)-Vertex-Coloring
cast classic LOCAL graph problems as hypergraph maximal matching problems (LOCAL reductions)
![Page 21: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/21.jpg)
(2Δ − 1)-Edge-Coloring
Maximal Independent Set
Unified Formulation as Hypergraph Maximal Matching Problem
Maximal Matching
(Δ + 1)-Vertex-Coloring
cast classic LOCAL graph problems as hypergraph maximal matching problems (LOCAL reductions)
rank 2 rank 3
rank Δ + 1rank Δ
![Page 22: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/22.jpg)
(2Δ − 1)-Edge-Coloring
Maximal Independent Set
Unified Formulation as Hypergraph Maximal Matching Problem
Maximal Matching
(Δ + 1)-Vertex-Coloring
cast classic LOCAL graph problems as hypergraph maximal matching problems (LOCAL reductions)
smooth interpolation between maximal matching and maximal independent set
rank 2 rank 3
rank Δ + 1rank Δ
![Page 23: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/23.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
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𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of Graph
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𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 26: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/26.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 27: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/27.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 28: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/28.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 29: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/29.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 30: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/30.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 31: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/31.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 32: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/32.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 33: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/33.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 34: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/34.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 35: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/35.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 36: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/36.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 37: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/37.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
![Page 38: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/38.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
properness
![Page 39: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/39.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
at most one color per edge
![Page 40: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/40.jpg)
𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching
Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching
at least one color per edge
![Page 41: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/41.jpg)
II) Hypergraph Maximal Matching Algorithm
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II) Hypergraph Maximal Matching Algorithm
𝑶(𝒓𝟐)-Approximate Maximum Matching
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II) Hypergraph Maximal Matching Algorithm
𝑶(𝒓𝟐)-Approximate Maximum Matching
𝑶 𝒓 -Approximate Maximum Fractional Matching
![Page 44: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/44.jpg)
II) Hypergraph Maximal Matching Algorithm
𝑶(𝒓𝟐)-Approximate Maximum Matching
𝑶 𝒓 -Approximate Maximum Fractional Matching
Rounding Fractional Matching
![Page 45: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/45.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
![Page 46: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/46.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
value of v
![Page 47: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/47.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
value of v
![Page 48: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/48.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
value of v
![Page 49: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/49.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
value of v
![Page 50: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/50.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 51: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/51.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 52: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/52.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
1
4
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 53: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/53.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
1
4
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 54: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/54.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
1
4
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 55: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/55.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
1
4
1
2
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 56: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/56.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
1
4
1
2
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 57: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/57.jpg)
Fractional Maximum Matching(2𝑟)-Approximate Fractional Matchingmax
𝑒∈𝐸
𝑥𝑒
s.t. for all 𝑣 ∈ 𝑉
𝑥𝑒 ∈ [0,1] for all 𝑒 ∈ 𝐸
𝑒∈𝐸(𝑣)
𝑥𝑒 ≤ 1
LOCAL Greedy Algorithm
𝑥𝑒 = 2−⌈log Δ⌉ for all 𝑒 ∈ 𝐸
repeat until all edges are blocked
mark half-tight nodes
block their edges
double value of unblocked edges
1
8
1
4
1
2
v is half-tight if its value is ≥𝟏
𝟐
value of v
![Page 58: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/58.jpg)
𝑶(𝒓𝟐)-Approximate Maximum Matching
𝑶 𝒓 -Approximate Maximum Fractional Matching
Rounding Fractional Matching
![Page 59: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/59.jpg)
Rounding Fractional Matching
![Page 60: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/60.jpg)
Rounding Fractional Matching
Ω1
Δ
1
![Page 61: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/61.jpg)
Rounding Fractional Matching
Direct Rounding
Ω1
Δ
1
![Page 62: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/62.jpg)
Rounding Fractional Matching
Direct Rounding
Iterative Rounding
Ω1
Δ
1
Ω1
Δ
1
![Page 63: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/63.jpg)
Rounding Fractional Matching
Direct Rounding
Iterative Rounding
Ω1
Δ
1
Ω1
Δ
1
![Page 64: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/64.jpg)
Factor-2-Rounding
![Page 65: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/65.jpg)
Factor-2-Rounding
Graphs:
factor-2-rounding with essentially no loss
approach used by Hańćkowiak, Karoński, Panconesi [SODA’98, PODC’99] and F. [DISC’17]
![Page 66: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/66.jpg)
Factor-2-Rounding
Graphs:
factor-2-rounding with essentially no loss
approach used by Hańćkowiak, Karoński, Panconesi [SODA’98, PODC’99] and F. [DISC’17]
![Page 67: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/67.jpg)
Factor-2-Rounding
Graphs:
factor-2-rounding with essentially no loss
approach used by Hańćkowiak, Karoński, Panconesi [SODA’98, PODC’99] and F. [DISC’17]
Hypergraphs?
Challenge: factor-2-rounding with almost no loss
![Page 68: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/68.jpg)
Factor-2-Rounding
Graphs:
factor-2-rounding with essentially no loss
approach used by Hańćkowiak, Karoński, Panconesi [SODA’98, PODC’99] and F. [DISC’17]
Hypergraphs?
Challenge: factor-2-rounding with almost no loss ?The Only Obstacle
Ghaffari, Kuhn, Maus [STOC’17]
![Page 69: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/69.jpg)
Factor-2-Rounding
Graphs:
factor-2-rounding with essentially no loss
approach used by Hańćkowiak, Karoński, Panconesi [SODA’98, PODC’99] and F. [DISC’17]
Challenge 1: factor-2-rounding with loss 𝑂(𝑟)
Hypergraphs?
Challenge: factor-2-rounding with almost no loss ?The Only Obstacle
Ghaffari, Kuhn, Maus [STOC’17]
![Page 70: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/70.jpg)
Factor-2-Rounding
Graphs:
factor-2-rounding with essentially no loss
approach used by Hańćkowiak, Karoński, Panconesi [SODA’98, PODC’99] and F. [DISC’17]
Challenge 1: factor-2-rounding with loss 𝑂(𝑟)
Hypergraphs?
Basic Rounding:factor-𝐿-rounding with loss 𝑂(𝑟)
Challenge: factor-2-rounding with almost no loss ?The Only Obstacle
Ghaffari, Kuhn, Maus [STOC’17]
![Page 71: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/71.jpg)
Basic Rounding
![Page 72: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/72.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 73: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/73.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 74: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/74.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 75: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/75.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 76: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/76.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 77: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/77.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 78: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/78.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 79: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/79.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 80: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/80.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 81: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/81.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 82: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/82.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 83: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/83.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 84: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/84.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 85: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/85.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 86: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/86.jpg)
Sequential Greedy Factor-2-rounding
for all unblocked edges with value <𝐿
𝑑
set value to𝐿
𝑑
mark tight nodes
block their edges
for all blocked edges with value <𝐿
𝑑
set value to 0
O 𝑟 loss
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 87: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/87.jpg)
LOCAL Greedy Factor-L-Rounding
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 88: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/88.jpg)
LOCAL Greedy Factor-L-Rounding
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 89: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/89.jpg)
LOCAL Greedy Factor-L-Rounding
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 90: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/90.jpg)
LOCAL Greedy Factor-L-Rounding
In each step, value of a node increased by at most +𝑑
2𝐿⋅𝐿
𝑑=
1
2
𝑂 log𝛥 + 𝐿 ⋅ 𝑟 2 rounds using Defective-Coloring Algorithm by Kuhn [SPAA’09]
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 91: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/91.jpg)
LOCAL Greedy Factor-L-Rounding
In each step, value of a node increased by at most +𝑑
2𝐿⋅𝐿
𝑑=
1
2
𝑂 log𝛥 + 𝐿 ⋅ 𝑟 2 rounds using Defective-Coloring Algorithm by Kuhn [SPAA’09]
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 92: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/92.jpg)
LOCAL Greedy Factor-L-Rounding
In each step, value of a node increased by at most +𝑑
2𝐿⋅𝐿
𝑑=
1
2
𝑂 log𝛥 + 𝐿 ⋅ 𝑟 2 rounds using Defective-Coloring Algorithm by Kuhn [SPAA’09]
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 93: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/93.jpg)
LOCAL Greedy Factor-L-Rounding
In each step, value of a node increased by at most +𝑑
2𝐿⋅𝐿
𝑑=
1
2
𝑂 log𝛥 + 𝐿 ⋅ 𝑟 2 rounds using Defective-Coloring Algorithm by Kuhn [SPAA’09]
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 94: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/94.jpg)
LOCAL Greedy Factor-L-Rounding
In each step, value of a node increased by at most +𝑑
2𝐿⋅𝐿
𝑑=
1
2
𝑂 log𝛥 + 𝐿 ⋅ 𝑟 2 rounds using Defective-Coloring Algorithm by Kuhn [SPAA’09]
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 95: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/95.jpg)
LOCAL Greedy Factor-L-Rounding
In each step, value of a node increased by at most +𝑑
2𝐿⋅𝐿
𝑑=
1
2
𝑂 log𝛥 + 𝐿 ⋅ 𝑟 2 rounds using Defective-Coloring Algorithm by Kuhn [SPAA’09]
𝑑
2𝐿- Defective 𝑂 𝐿2𝑟2 -Edge-Coloring
for each color class
mark half-tight nodes
block their edges
set value of edges in color class to𝐿
𝑑
for all blocked edges with value <𝐿
𝑑
set value to 0
Basic Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
from ≥1
𝑑to ≥
𝐿
𝑑
![Page 96: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/96.jpg)
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 lossBasic Rounding:
![Page 97: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/97.jpg)
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds 𝐿 = O(Δ)
Basic Rounding:
1
![Page 98: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/98.jpg)
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
𝐿 = O(Δ)
Basic Rounding:
1
![Page 99: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/99.jpg)
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 100: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/100.jpg)
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 101: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/101.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 102: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/102.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 103: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/103.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 104: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/104.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 105: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/105.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 106: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/106.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 107: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/107.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding:
1
Ω1
Δ1
![Page 108: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/108.jpg)
Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding
Basic Rounding:
1
Ω1
Δ1
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Challenge 2: repeated application
Ω1
Δ
Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 loss
Direct Basic Rounding
𝑂 𝑟 loss
𝑂 logΔ + Δ ⋅ 𝑟 2 rounds
too slow!
Iterative Basic Rounding
𝑟Θ(log Δ) loss
𝑂 logΔ ⋅ log Δ + 𝑟2 rounds
too lossy!
Recursive Rounding&
Iterative Refilling
𝐿 = O(Δ)
𝐿 = O(1)
Basic Rounding
Basic Rounding:
Invariant: 𝑶(𝒓) loss, kept using refilling iterations
1
Ω1
Δ1
![Page 110: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/110.jpg)
Recursive Round
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Recursive Round
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Recursive Round
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Recursive Round
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Recursive Round
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Recursive Round
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Recursive Round
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Recursive Round
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Recursive Round
round 𝑳
O(r) loss
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Recursive Round
round 𝑳
O(r) loss
round 𝑳
O(r) loss
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Recursive Round
round 𝑳
O(r) loss
round 𝑳
O(r) loss
round’(L) with O 𝒓𝟐 loss
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Iterative Refill
![Page 122: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/122.jpg)
round’(L)O 𝑟2 loss
Iterative Refill
![Page 123: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/123.jpg)
round’(L)O 𝑟2 loss
Iterative Refill
subtract
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
ITERATIVE REFILLING
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round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
After O(r) refilling iterations:factor-L-roundingO(r) loss
ITERATIVE REFILLING
![Page 131: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/131.jpg)
round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
After O(r) refilling iterations:factor-L-roundingO(r) loss
ITERATIVE REFILLING
round(L)
![Page 132: Deterministic Distributed Edge-Coloring - ETH Zpeople.inf.ethz.ch/fiscmanu/talk/edgecol.pdf · Deterministic Distributed Edge-Coloring via ... deterministic algorithm for Maximal](https://reader030.fdocument.org/reader030/viewer/2022021507/5b0659fe7f8b9abf568ce849/html5/thumbnails/132.jpg)
round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
After O(r) refilling iterations:factor-L-roundingO(r) loss
ITERATIVE REFILLING
round(L)
𝑻 𝑳 = 𝑶 𝒓 ⋅ 𝑻′ 𝑳 = 𝑶 𝒓 ⋅ 𝑻 𝑳
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𝑻 𝑶(𝟏) = 𝐥𝐨𝐠𝜟 + 𝑶 𝒓𝟐
round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
After O(r) refilling iterations:factor-L-roundingO(r) loss
ITERATIVE REFILLING
round(L)
𝑻 𝑳 = 𝑶 𝒓 ⋅ 𝑻′ 𝑳 = 𝑶 𝒓 ⋅ 𝑻 𝑳
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𝑻 𝑶(𝟏) = 𝐥𝐨𝐠𝜟 + 𝑶 𝒓𝟐
round’(L)O 𝑟2 loss
Iterative Refill
subtract
round’(L)O 𝑟2 loss
add
After O(r) refilling iterations:factor-L-roundingO(r) loss
ITERATIVE REFILLING
round(L)
𝑻 𝑳 = 𝑶 𝒓 ⋅ 𝑻′ 𝑳 = 𝑶 𝒓 ⋅ 𝑻 𝑳𝑻 𝑶(𝜟) = 𝒓𝑶(𝐥𝐨𝐠 𝐥𝐨𝐠 𝜟) ⋅ 𝐥𝐨𝐠 𝜟 + 𝑶 𝒓𝟐
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Conclusion & Open Problems
• (𝟐𝜟 − 𝟏)-Edge-Coloring is efficient: 𝑶(𝐥𝐨𝐠𝟖 𝒏)
• Linial’s Question from the 1980s:
Is there an efficient deterministic algorithm for Maximal Independent Set?
• Generality of Hypergraph Maximal Matchings
• Devise or rule out a poly 𝑟 ⋅ log 𝑛 -round deterministic algorithm
for rank-r-hypergraph maximal matching.
• Key Problem: Efficient Deterministic Rounding
• Completeness of Rounding (Ghaffari, Kuhn, Maus [STOC’17])
Rounding as the only obstacle for efficient deterministic LOCAL graph algorithms
• Devise a more general deterministic rounding method.
𝐩𝐨𝐥𝐲 𝐫 ⋅ 𝐥𝐨𝐠𝐎 𝐥𝐨𝐠 𝒓 𝜟 ⋅ 𝐥𝐨𝐠 𝐧 rounds