SlideShare a Scribd company logo

slidespdf

•
•
A
aman gour

The document discusses conflict-free coloring of graphs. It first defines conflict-free coloring as assigning colors to vertices such that each vertex has a uniquely colored neighbor. It then summarizes previous work showing that conflict-free coloring planar graphs with 1 or 2 colors is NP-complete, while coloring general graphs with any number of colors is NP-complete. The document aims to determine the minimum number of colors needed for conflict-free coloring planar graphs.

1 of 94
Download to read offline
Aman Gour, Phillip Keldenich Zachery Abel, Victor Alvarez, Erik Demaine, Sándor Fekete, Adam Hesterberg, Christian Scheffer Three Colors Suffice: Conflict-free Coloring of Planar Graphs
Planar Four-Color Theorem Planar four-color theorem:
 Every planar graph is 4-colorable. 2 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. Partial results: k < 4: Easy to see k = 4: 4-color theorem k < 7: Robertson et al. (1993) k = 7 and higher: Open 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Preface: Conflict-Free Coloring 4
Conflict-Free Coloring Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Conflict-Free Coloring: Questions 7 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs • General graphs
Conflict-Free Coloring: Questions 7 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs • General graphs Sufficient number of colors • Planar graphs • General graphs
Conflict-Free Coloring: Questions 7 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs • General graphs Sufficient number of colors • Planar graphs • General graphs Number of colored vertices — Planar graphs
Conflict-Free Coloring: Answers 8 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs: NP-complete for 1 and 2 colors • General graphs: NP-complete for any number of colors
Conflict-Free Coloring: Answers 8 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs: NP-complete for 1 and 2 colors • General graphs: NP-complete for any number of colors Sufficient number of colors • Three colors suffice for planar graphs • Conflict-free analogue of Hadwiger’s Conjecture
Conflict-Free Coloring: Answers 8 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs: NP-complete for 1 and 2 colors • General graphs: NP-complete for any number of colors Sufficient number of colors • Three colors suffice for planar graphs • Conflict-free analogue of Hadwiger’s Conjecture Number of colored vertices - Planar graphs • k = 3: No constant approximation factor • k = 4: FPT, PTAS, dom(G) always possible
Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Part I: Complexity 9
Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4
Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 c1 c1;1 c1;3c1
Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 x1 c1 c1;1 c1;3c1
Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 z1;1 c1;1 c1;3 z1;3 x1 x2 x3 x4 x5 c1 G1(Φ)
Complexity in Planar Graphs 11 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Proof: Reduction from planar conflict-free 1-coloring
Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 v
From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 z1;1 c1;1 c1;3 z1;3 x1 x2 x3 x4 x5 c1
From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f
From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f G2(Φ)
From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f G2(Φ)
From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f G2(Φ)
Complexity for General Graphs 14 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Proof: Reduction from proper coloring
Complexity for General Graphs 14 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Proof: Reduction from proper coloring Basic Idea: Gadgets enforcing vertices in a graph to be colored and edges to be colored with distinct colors at both end points. Gk Gk . . . . . .. . . w Gk 1 Gk 1 . . . Gk Gk v
Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Part II: Sufficient number of colors 15
Sufficient Criterion 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Sufficient Criterion K 3 5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Sufficient Criterion K 3 5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Sufficient Criterion K 3 5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Sufficient Criterion K 3 5 K 3 6 K5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
Proof 18 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof By induction: For k=1: Collection of paths K 3 4 18 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof By induction: For k=1: Collection of paths Induction step: Removal of distance-3-set reduces k by one 19 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof Induction step: Removal of distance-3-set reduces k by one Unseen vertices U 20 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof Claim: No set U of unseen vertices is a cutset of G. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
Proof Claim: No set U of unseen vertices is a cutset of G. Hv0 ≥ 2 ≥ 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
Proof Claim: No set U of unseen vertices is a cutset of G. Hv0 ≥ 2 ≥ 2 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
Proof Claim: No set U of unseen vertices is a cutset of G. Hw0 w−1 = 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
Proof Claim: No set U of unseen vertices is a cutset of G. Hw0 w−1 = 3 = 2 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Uv0 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Uv0 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Kk+1 U v 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Kk+1 U v Paths could be contracted, yielding a Kk+2 minor! 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Kk+1 U v Paths could be contracted, yielding a Kk+2 minor! Other case analogous - we are done! 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Future Work 23
Future Work Neighborhoods: Open versus closed neighborhoods 24 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Neighborhoods: Open versus closed neighborhoods Other graph classes:
 Intersection graphs of geometric objects 26 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Neighborhoods: Open versus closed neighborhoods Other graph classes:
 Intersection graphs of geometric objects 26 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
Future Work Conflict-free AGP:
 Visibility graphs of polygons, point sets in polygons 27 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Neighborhoods: Open versus closed neighborhoods Other graph classes:
 Intersection graphs of geometric objects
Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Gracias! 28

Recommended

Bipartite graph
Bipartite graphBipartite graph
Bipartite graph
Arafat Hossan
 
Algorithm to count number of disjoint paths
Algorithm to count number of disjoint pathsAlgorithm to count number of disjoint paths
Algorithm to count number of disjoint paths
Sujith Jay Nair
 
Maths Games & Videos
Maths Games & VideosMaths Games & Videos
Maths Games & Videos
Suhaila
 
Adding Fractions Mathematics GCSE Maths
Adding Fractions Mathematics GCSE MathsAdding Fractions Mathematics GCSE Maths
Adding Fractions Mathematics GCSE Maths
How2Become.com
 
Healthy Living and Healthy Eating EDU 290 Powerpoint
Healthy Living and Healthy Eating EDU 290 PowerpointHealthy Living and Healthy Eating EDU 290 Powerpoint
Healthy Living and Healthy Eating EDU 290 Powerpoint
Kris Barron
 
How The Love of Music has changed our Business World
How The Love of Music has changed our Business WorldHow The Love of Music has changed our Business World
How The Love of Music has changed our Business World
Thorsten Faltings
 
Blazing Performance with Flame Graphs
Blazing Performance with Flame GraphsBlazing Performance with Flame Graphs
Blazing Performance with Flame Graphs
Brendan Gregg
 
Powerpoint on class rules
Powerpoint on class rulesPowerpoint on class rules
Powerpoint on class rules
Shodge2
 

Recommended

Bipartite graph
Bipartite graphBipartite graph
Bipartite graph
Arafat Hossan
 
Algorithm to count number of disjoint paths
Algorithm to count number of disjoint pathsAlgorithm to count number of disjoint paths
Algorithm to count number of disjoint paths
Sujith Jay Nair
 
Maths Games & Videos
Maths Games & VideosMaths Games & Videos
Maths Games & Videos
Suhaila
 
Adding Fractions Mathematics GCSE Maths
Adding Fractions Mathematics GCSE MathsAdding Fractions Mathematics GCSE Maths
Adding Fractions Mathematics GCSE Maths
How2Become.com
 
Healthy Living and Healthy Eating EDU 290 Powerpoint
Healthy Living and Healthy Eating EDU 290 PowerpointHealthy Living and Healthy Eating EDU 290 Powerpoint
Healthy Living and Healthy Eating EDU 290 Powerpoint
Kris Barron
 
How The Love of Music has changed our Business World
How The Love of Music has changed our Business WorldHow The Love of Music has changed our Business World
How The Love of Music has changed our Business World
Thorsten Faltings
 
Blazing Performance with Flame Graphs
Blazing Performance with Flame GraphsBlazing Performance with Flame Graphs
Blazing Performance with Flame Graphs
Brendan Gregg
 
Powerpoint on class rules
Powerpoint on class rulesPowerpoint on class rules
Powerpoint on class rules
Shodge2
 
Lecture-9.pptx
Lecture-9.pptxLecture-9.pptx
Lecture-9.pptx
ssuserf67e3a
 
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple GraphsGreedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
inventionjournals
 
Edge Coloring & K-tuple coloring
Edge Coloring & K-tuple coloringEdge Coloring & K-tuple coloring
Edge Coloring & K-tuple coloring
Dr. Abdul Ahad Abro
 
GRAPH COLORING AND ITS APPLICATIONS
GRAPH COLORING AND ITS APPLICATIONSGRAPH COLORING AND ITS APPLICATIONS
GRAPH COLORING AND ITS APPLICATIONS
Manojit Chakraborty
 
A study-of-vertex-edge-coloring-techniques-with-application
A study-of-vertex-edge-coloring-techniques-with-applicationA study-of-vertex-edge-coloring-techniques-with-application
A study-of-vertex-edge-coloring-techniques-with-application
Ijcem Journal
 
Graph coloring
Graph coloringGraph coloring
Graph coloring
Barani Tharan
 
Graph coloring problem
Graph coloring problemGraph coloring problem
Graph coloring problem
V.V.Vanniaperumal College for Women
 
Graph Coloring
Graph ColoringGraph Coloring
Graph Coloring
Dr. Abdul Ahad Abro
 

More Related Content

Similar to slidespdf

Lecture-9.pptx
Lecture-9.pptxLecture-9.pptx
Lecture-9.pptx
ssuserf67e3a
 
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple GraphsGreedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
inventionjournals
 
Edge Coloring & K-tuple coloring
Edge Coloring & K-tuple coloringEdge Coloring & K-tuple coloring
Edge Coloring & K-tuple coloring
Dr. Abdul Ahad Abro
 
GRAPH COLORING AND ITS APPLICATIONS
GRAPH COLORING AND ITS APPLICATIONSGRAPH COLORING AND ITS APPLICATIONS
GRAPH COLORING AND ITS APPLICATIONS
Manojit Chakraborty
 
A study-of-vertex-edge-coloring-techniques-with-application
A study-of-vertex-edge-coloring-techniques-with-applicationA study-of-vertex-edge-coloring-techniques-with-application
A study-of-vertex-edge-coloring-techniques-with-application
Ijcem Journal
 
Graph coloring
Graph coloringGraph coloring
Graph coloring
Barani Tharan
 
Graph coloring problem
Graph coloring problemGraph coloring problem
Graph coloring problem
V.V.Vanniaperumal College for Women
 
Graph Coloring
Graph ColoringGraph Coloring
Graph Coloring
Dr. Abdul Ahad Abro
 

Similar to slidespdf (8)

Lecture-9.pptx
Lecture-9.pptxLecture-9.pptx
Lecture-9.pptx
 
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple GraphsGreedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphs
 
Edge Coloring & K-tuple coloring
Edge Coloring & K-tuple coloringEdge Coloring & K-tuple coloring
Edge Coloring & K-tuple coloring
 
GRAPH COLORING AND ITS APPLICATIONS
GRAPH COLORING AND ITS APPLICATIONSGRAPH COLORING AND ITS APPLICATIONS
GRAPH COLORING AND ITS APPLICATIONS
 
A study-of-vertex-edge-coloring-techniques-with-application
A study-of-vertex-edge-coloring-techniques-with-applicationA study-of-vertex-edge-coloring-techniques-with-application
A study-of-vertex-edge-coloring-techniques-with-application
 
Graph coloring
Graph coloringGraph coloring
Graph coloring
 
Graph coloring problem
Graph coloring problemGraph coloring problem
Graph coloring problem
 
Graph Coloring
Graph ColoringGraph Coloring
Graph Coloring
 

slidespdf

  • 1. Aman Gour, Phillip Keldenich Zachery Abel, Victor Alvarez, Erik Demaine, Sándor Fekete, Adam Hesterberg, Christian Scheffer Three Colors Suffice: Conflict-free Coloring of Planar Graphs
  • 2. Planar Four-Color Theorem Planar four-color theorem:
 Every planar graph is 4-colorable. 2 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 3. Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 4. Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 5. Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 6. Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 7. Hadwiger’s Conjecture Hugo Hadwiger (1943): Every graph G with chromatic number χ(G) = k has Hadwiger number H(G) ≥ k, i.e., it has Kk, the complete graph on k vertices, as a minor. Partial results: k < 4: Easy to see k = 4: 4-color theorem k < 7: Robertson et al. (1993) k = 7 and higher: Open 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 8. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Preface: Conflict-Free Coloring 4
  • 9. Conflict-Free Coloring Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 10. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 11. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 12. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 13. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 14. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 15. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 16. Conflict-Free Coloring v u1 u2 u3 u4 u5 u6 u7 u8 Coloring of a subset of the vertices Conflict-free coloring: such that every vertex v has a conflict-free neighbor u 2 N[v] whose color appears in N[v] exactly once. 5 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 17. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 18. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 19. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 20. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 21. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 22. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 23. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 24. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 25. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 26. Conflict-Free Coloring 6 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 27. Conflict-Free Coloring: Questions 7 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs • General graphs
  • 28. Conflict-Free Coloring: Questions 7 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs • General graphs Sufficient number of colors • Planar graphs • General graphs
  • 29. Conflict-Free Coloring: Questions 7 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs • General graphs Sufficient number of colors • Planar graphs • General graphs Number of colored vertices — Planar graphs
  • 30. Conflict-Free Coloring: Answers 8 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs: NP-complete for 1 and 2 colors • General graphs: NP-complete for any number of colors
  • 31. Conflict-Free Coloring: Answers 8 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs: NP-complete for 1 and 2 colors • General graphs: NP-complete for any number of colors Sufficient number of colors • Three colors suffice for planar graphs • Conflict-free analogue of Hadwiger’s Conjecture
  • 32. Conflict-Free Coloring: Answers 8 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Complexity • Planar graphs: NP-complete for 1 and 2 colors • General graphs: NP-complete for any number of colors Sufficient number of colors • Three colors suffice for planar graphs • Conflict-free analogue of Hadwiger’s Conjecture Number of colored vertices - Planar graphs • k = 3: No constant approximation factor • k = 4: FPT, PTAS, dom(G) always possible
  • 33. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Part I: Complexity 9
  • 34. Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 35. Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4
  • 36. Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 c1 c1;1 c1;3c1
  • 37. Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 x1 c1 c1;1 c1;3c1
  • 38. Complexity in Planar Graphs 10 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 z1;1 c1;1 c1;3 z1;3 x1 x2 x3 x4 x5 c1 G1(Φ)
  • 39. Complexity in Planar Graphs 11 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Proof: Reduction from planar conflict-free 1-coloring
  • 40. Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 41. Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 42. Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 43. Gadget G≤1 Idea: G≤1 reduces the number of colors available to 1 12 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 v
  • 44. From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 45. From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 x1 x2 x3 x4 x5 c1 c2 c3 c4 z1;1 c1;1 c1;3 z1;3 x1 x2 x3 x4 x5 c1
  • 46. From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f
  • 47. From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f G2(Φ)
  • 48. From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f G2(Φ)
  • 49. From 2 Colors to 1 Color 13 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Zi ≤ 1 Zi+1≤ 1 ≤ 1 cj;3 cj;1 ≤ 1 cj Zi−1 upper lower left right t f f G2(Φ)
  • 50. Complexity for General Graphs 14 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Proof: Reduction from proper coloring
  • 51. Complexity for General Graphs 14 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Proof: Reduction from proper coloring Basic Idea: Gadgets enforcing vertices in a graph to be colored and edges to be colored with distinct colors at both end points. Gk Gk . . . . . .. . . w Gk 1 Gk 1 . . . Gk Gk v
  • 52. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Part II: Sufficient number of colors 15
  • 53. Sufficient Criterion 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 54. Sufficient Criterion K 3 5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 55. Sufficient Criterion K 3 5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 56. Sufficient Criterion K 3 5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 57. Sufficient Criterion K 3 5 K 3 6 K5 16 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 58. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 59. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 60. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 61. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 62. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 63. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 64. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 65. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 66. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 67. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 68. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 69. Coloring Algorithm 17 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 • Iteratively construct color class (distance-3-set) • Always choose vertices of distance exactly 3 • Remove covered vertices • Repeat until what remains is a collection of paths
  • 70. Proof 18 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 71. Proof By induction: For k=1: Collection of paths K 3 4 18 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 72. Proof By induction: For k=1: Collection of paths Induction step: Removal of distance-3-set reduces k by one 19 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 73. Proof Induction step: Removal of distance-3-set reduces k by one Unseen vertices U 20 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 74. Proof Claim: No set U of unseen vertices is a cutset of G. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
  • 75. Proof Claim: No set U of unseen vertices is a cutset of G. Hv0 ≥ 2 ≥ 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
  • 76. Proof Claim: No set U of unseen vertices is a cutset of G. Hv0 ≥ 2 ≥ 2 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
  • 77. Proof Claim: No set U of unseen vertices is a cutset of G. Hw0 w−1 = 3 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
  • 78. Proof Claim: No set U of unseen vertices is a cutset of G. Hw0 w−1 = 3 = 2 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 21
  • 79. Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 80. Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Uv0 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 81. Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Uv0 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 82. Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Kk+1 U v 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 83. Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Kk+1 U v Paths could be contracted, yielding a Kk+2 minor! 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 84. Proof G[U] does not contain a Kk+1 or Kk+2 as minor-3 Kk+1 U v Paths could be contracted, yielding a Kk+2 minor! Other case analogous - we are done! 22 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 85. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Future Work 23
  • 86. Future Work Neighborhoods: Open versus closed neighborhoods 24 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 87. Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 88. Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 89. Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 90. Future Work Neighborhoods: Open versus closed neighborhoods 25 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 91. Future Work Neighborhoods: Open versus closed neighborhoods Other graph classes:
 Intersection graphs of geometric objects 26 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 92. Future Work Neighborhoods: Open versus closed neighborhoods Other graph classes:
 Intersection graphs of geometric objects 26 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017
  • 93. Future Work Conflict-free AGP:
 Visibility graphs of polygons, point sets in polygons 27 Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Neighborhoods: Open versus closed neighborhoods Other graph classes:
 Intersection graphs of geometric objects
  • 94. Aman Gour | Phillip Keldenich | Three Colors Suffice: Conflict-Free Coloring of Planar Graphs | SODA 2017 Gracias! 28