Post on 24-May-2018
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