Parameterized Complexity of Conflictfree Graph Coloring WADS 2019

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Parameterized Complexity of Conflict-free Graph Coloring WADS 2019 Hans L. Bodlaender 1, Sudeshna Kolay

Parameterized Complexity of Conflict-free Graph Coloring WADS 2019 Hans L. Bodlaender 1, Sudeshna Kolay 2, and Astrid Pieterse 3 1 Utrecht University, The Netherlands 2 Ben Gurion University of Negev, Israel 3 Eindhoven University of Technology, The Netherlands

Conflict-free graph coloring • z y u v w x

Conflict-free graph coloring • z y u v w x

CNCF-Coloring • 2 u v w y x

CNCF-Coloring • 2 u v w y x

CNCF-Coloring • 2 2 w u v x y

CNCF-Coloring • 2 2 w u v x y

CNCF-Coloring • 2 2 2 u v w y x

CNCF-Coloring • 2 2 2 u v w y x

CNCF-Coloring • 2 2

CNCF-Coloring • 2 2

CNCF-Coloring • 2 2 k Observation Any proper coloring is a CNCF-Coloring • Each

CNCF-Coloring • 2 2 k Observation Any proper coloring is a CNCF-Coloring • Each vertex is the uniquely colored vertex in its closed neighborhood

Background •

Background •

Combinatorial results • … …

Combinatorial results • … …

Background •

Background •

Algorithmic results •

Algorithmic results •

Kernelization results • Efficie nt Small bound Corre c t

Kernelization results • Efficie nt Small bound Corre c t

A general reduction rule

A general reduction rule

Reducing number of twins • s

Reducing number of twins • s

Correctness • s s

Correctness • s s

Correctness • s s

Correctness • s s

Effectiveness •

Effectiveness •

Polynomial kernel for 2 -CNCF-Coloring Parameterized by Vertex Cover size

Polynomial kernel for 2 -CNCF-Coloring Parameterized by Vertex Cover size

A generalized kernel •

A generalized kernel •

Kernelization: General idea •

Kernelization: General idea •

Removing low-degree vertices •

Removing low-degree vertices •

Removing low-degree vertices •

Removing low-degree vertices •

Rewriting: Basics •

Rewriting: Basics •

Rewriting: Continued •

Rewriting: Continued •

Reducing the number of variables • S S S S

Reducing the number of variables • S S S S

Reducing the number of variables •

Reducing the number of variables •

Resulting Poly root CSP instance •

Resulting Poly root CSP instance •

Kernelization •

Kernelization •

Conclusion •

Conclusion •

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