Parameterized Pre-Coloring Extension and List Coloring Problems - PowerPoint PPT Presentation

Parameterized Pre-Coloring Extension and List Coloring Problems Gregory Gutin 1 Diptapriyo Majumdar 1 Sebastian Ordyniak 2 Magnus Wahlström 1 1 Royal Holloway, University of London, United Kingdom 2 University of Sheffield, United Kingdom April 8, 2020, BCTCS, Swansea, United Kingdom 1

Outline 1 Definition and Properties 2 Our Results 3 Pre-Coloring Extension 4 Conclusions 2

Parameterized Problem • A parameterized problem is a language L ⊆ Σ ∗ × N . Input instance of L is ( x, k ) where x ∈ Σ ∗ , k ∈ N . k is called parameter . 3

Parameterized Problem • A parameterized problem is a language L ⊆ Σ ∗ × N . Input instance of L is ( x, k ) where x ∈ Σ ∗ , k ∈ N . k is called parameter . • Example: Vertex Cover parameterized by Solution Size. L = { ( G, k ) |∃ S ⊆ V ( G ) such that | S | ≤ k and G \ S has no edge } . 3

Fixed-Parameter Tractability (FPT) Yes if ( x, k ) 2 L A ( x, k ) No , otherwise • Algorithm A runs in f ( k ) · | x | c time. • A is called Fixed Parameter Algorithm . 4

Hardness in Parameterized Complexity 5

Hardness in Parameterized Complexity FPT ⊆ W[1] ⊆ W[2] ⊆ . . . ⊆ XP. 5

Kernelization ( x 0 , k 0 ) Preprocess ( x, k ) • Preprocessing takes poly ( | x | , k ) time. • ( x, k ) 2 L if and only if ( x 0 , k 0 ) 2 L . • | x 0 | + k 0 ≤ g ( k ). • If g ( k ) = poly ( k ), then we say that L has a polynomial kernel. 6

Graph Coloring 7

Graph Coloring p -Coloring Input: An undirected graph G = ( V, E ) and a set of p colors Q . Goal: Does there exist λ : V ( G ) → Q such that for every u, v ∈ V ( G ) , λ ( u ) � = λ ( v ) ? 7

Graph Coloring p -Coloring Input: An undirected graph G = ( V, E ) and a set of p colors Q . Goal: Does there exist λ : V ( G ) → Q such that for every u, v ∈ V ( G ) , λ ( u ) � = λ ( v ) ? • For p ≤ 2 , p -Coloring is polynomial time solvable. 7

Graph Coloring p -Coloring Input: An undirected graph G = ( V, E ) and a set of p colors Q . Goal: Does there exist λ : V ( G ) → Q such that for every u, v ∈ V ( G ) , λ ( u ) � = λ ( v ) ? • For p ≤ 2 , p -Coloring is polynomial time solvable. • For p ≥ 3 , p -Coloring is NP-Complete in general graphs. 7

Graph Coloring p -Coloring Input: An undirected graph G = ( V, E ) and a set of p colors Q . Goal: Does there exist λ : V ( G ) → Q such that for every u, v ∈ V ( G ) , λ ( u ) � = λ ( v ) ? • For p ≤ 2 , p -Coloring is polynomial time solvable. • For p ≥ 3 , p -Coloring is NP-Complete in general graphs. • p -Coloring is polynomial time solvable on chordal graphs. 7

Precoloring Extension Pre-coloring Extension Input: A graph G , and a precoloring λ P : X → Q for X ⊆ V ( G ) where Q is a set of colors. Goal: Can λ P be extended to a proper coloring of G using colors from only Q ? 8

Precoloring Extension Pre-coloring Extension Input: A graph G , and a precoloring λ P : X → Q for X ⊆ V ( G ) where Q is a set of colors. Goal: Can λ P be extended to a proper coloring of G using colors from only Q ? • Pre-coloring Extension is polynomial time solvable in cluster graphs, but NP-Complete in bipartite graphs. 8

List Coloring List Coloring Input: A graph G , and a list L ( v ) for every v ∈ V ( G ) . Goal: Is there a proper coloring λ : V ( G ) → � L ( u ) such u ∈ V ( G ) that for every u ∈ V ( G ) , λ ( u ) ∈ L ( u ) ? 9

List Coloring List Coloring Input: A graph G , and a list L ( v ) for every v ∈ V ( G ) . Goal: Is there a proper coloring λ : V ( G ) → � L ( u ) such u ∈ V ( G ) that for every u ∈ V ( G ) , λ ( u ) ∈ L ( u ) ? • List Coloring is polynomial time solvable in clique, and cluster graph. 9

List Coloring List Coloring Input: A graph G , and a list L ( v ) for every v ∈ V ( G ) . Goal: Is there a proper coloring λ : V ( G ) → � L ( u ) such u ∈ V ( G ) that for every u ∈ V ( G ) , λ ( u ) ∈ L ( u ) ? • List Coloring is polynomial time solvable in clique, and cluster graph. • List Coloring is NP-Complete in split graphs, and graphs of cliquewidth two. 9

Outline 1 Definition and Properties 2 Our Results 3 Pre-Coloring Extension 4 Conclusions 10

Our problems and results 11

Our problems and results Pre-Coloring Extension Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, and a precoloring λ P : X �→ Q for X ⊆ V ( G ) where Q is a set of colors. Parameter: k Question: Can λ P be extended to a proper coloring of G using colors from only Q ? 11

Our problems and results Pre-Coloring Extension Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, and a precoloring λ P : X �→ Q for X ⊆ V ( G ) where Q is a set of colors. Parameter: k Question: Can λ P be extended to a proper coloring of G using colors from only Q ? List Coloring Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, a list L ( v ) of colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? 11

What is known From Paulusma (WG 2015) Parameter Coloring Pre-Color Ext List-Color clique-width W[1]-hard para-NPC para-NPC treewidth FPT W[1]-hard W[1]-hard cluster deletion FPT W[1]-hard W[1]-hard vertex cover FPT FPT W[1]-hard 12

Pre-coloring Extension 13

Pre-coloring Extension Pre-Coloring Extension Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, and a precoloring λ P : X �→ Q for X ⊆ V ( G ) where Q is a set of colors. Parameter: k Question: Can λ P be extended to a proper coloring of G using colors from only Q ? 13

Pre-coloring Extension Pre-Coloring Extension Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, and a precoloring λ P : X �→ Q for X ⊆ V ( G ) where Q is a set of colors. Parameter: k Question: Can λ P be extended to a proper coloring of G using colors from only Q ? • Golovach, Paulusma, and Song (2014) asked to determine the parameterized complexity status of Pre-Coloring Extension Clique Modulator . 13

Pre-coloring Extension Pre-Coloring Extension Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, and a precoloring λ P : X �→ Q for X ⊆ V ( G ) where Q is a set of colors. Parameter: k Question: Can λ P be extended to a proper coloring of G using colors from only Q ? • Golovach, Paulusma, and Song (2014) asked to determine the parameterized complexity status of Pre-Coloring Extension Clique Modulator . • We prove positively that Pre-Coloring Extension Clique Modulator is FPT. 13

Pre-coloring Extension Pre-Coloring Extension Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, and a precoloring λ P : X �→ Q for X ⊆ V ( G ) where Q is a set of colors. Parameter: k Question: Can λ P be extended to a proper coloring of G using colors from only Q ? • Golovach, Paulusma, and Song (2014) asked to determine the parameterized complexity status of Pre-Coloring Extension Clique Modulator . • We prove positively that Pre-Coloring Extension Clique Modulator is FPT. • We prove that Pre-Coloring Extension Clique Modulator admits a kernel with 3 k vertices. 13

List Coloring ( n − k ) -Regular List Coloring Input: A graph G , a list L ( v ) of ( n − k ) colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? 14

List Coloring ( n − k ) -Regular List Coloring Input: A graph G , a list L ( v ) of ( n − k ) colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? List Coloring Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, a list L ( v ) of colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? 14

List Coloring ( n − k ) -Regular List Coloring Input: A graph G , a list L ( v ) of ( n − k ) colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? List Coloring Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, a list L ( v ) of colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? • An instance ( G, L, k ) of ( n − k ) -Regular List Coloring can be transformed into an equivalent instance ( G ′ , D, L, k ′ ) of ( n − k ) -Regular List Coloring such that k ′ = 2 k . 14

List Coloring List Coloring Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, a list L ( v ) of colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? 15

List Coloring List Coloring Clique Modulator Input: A graph G , a clique modulator D with at most k ver- tices, a list L ( v ) of colors for every v ∈ V ( G ) . Parameter: k Question: Is there a proper list coloring of G ? • Golovach, Paulusma, and Song [2014] asked to determine the parameterized complexity status of this problem. 15