Deterministic Distributed Edge-Coloring via

Hypergraph Maximal Matching

Manuela FischerETH Zurich

Mohsen GhaffariETH Zurich

Fabian KuhnUni Freiburg

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

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

Maximal Independent Set

Classic LOCAL Graph Problems

Maximal Matching

(Δ + 1)-Vertex-Coloring

(2Δ − 1)-Edge-Coloring

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]

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]

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]

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]

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?

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?

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

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

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

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

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

I) Formulation as Hypergraph Maximal Matching

I) Formulation as Hypergraph Maximal Matching

II) Hypergraph Maximal Matching Algorithm

I) Formulation as Hypergraph Maximal Matching

(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

(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)

(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

(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

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of Graph

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

properness

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

at most one color per edge

𝟐𝚫 − 𝟏 -Edge-Coloring as Rank-3-Hypergraph Maximal Matching

Edge-Coloring of GraphRank-3-Hypergraph Maximal Matching

at least one color per edge

II) Hypergraph Maximal Matching Algorithm

II) Hypergraph Maximal Matching Algorithm

𝑶(𝒓𝟐)-Approximate Maximum Matching

II) Hypergraph Maximal Matching Algorithm

𝑶(𝒓𝟐)-Approximate Maximum Matching

𝑶 𝒓 -Approximate Maximum Fractional Matching

II) Hypergraph Maximal Matching Algorithm

𝑶(𝒓𝟐)-Approximate Maximum Matching

𝑶 𝒓 -Approximate Maximum Fractional Matching

Rounding Fractional Matching

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

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

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

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

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

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

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

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

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

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

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

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

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

𝑶(𝒓𝟐)-Approximate Maximum Matching

𝑶 𝒓 -Approximate Maximum Fractional Matching

Rounding Fractional Matching

Rounding Fractional Matching

Rounding Fractional Matching

Ω1

Δ

1

Rounding Fractional Matching

Direct Rounding

Ω1

Δ

1

Rounding Fractional Matching

Direct Rounding

Iterative Rounding

Ω1

Δ

1

Ω1

Δ

1

Rounding Fractional Matching

Direct Rounding

Iterative Rounding

Ω1

Δ

1

Ω1

Δ

1

Factor-2-Rounding

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]

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]

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

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]

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]

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]

Basic Rounding

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Rounding Factor-L-Rounding in 𝑂 log 𝛥 + 𝐿 ⋅ 𝑟 2 rounds with 𝑂 𝑟 lossBasic Rounding:

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

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

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

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

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

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

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

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

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

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

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

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

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

Recursive Round

Recursive Round

Recursive Round

Recursive Round

Recursive Round

Recursive Round

Recursive Round

Recursive Round

Recursive Round

round 𝑳

O(r) loss

Recursive Round

round 𝑳

O(r) loss

round 𝑳

O(r) loss

Recursive Round

round 𝑳

O(r) loss

round 𝑳

O(r) loss

round’(L) with O 𝒓𝟐 loss

Iterative Refill

round’(L)O 𝑟2 loss

Iterative Refill

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

add

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

add

round’(L)O 𝑟2 loss

Iterative Refill

subtract

round’(L)O 𝑟2 loss

add

ITERATIVE REFILLING

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)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)

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

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

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

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