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

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

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

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Parameterized Pre-Coloring Extension and List Coloring Problems

Gregory Gutin1 Diptapriyo Majumdar1 Sebastian Ordyniak2 Magnus Wahlström1

1Royal Holloway, University of London, United Kingdom 2University of Sheffield, United Kingdom

April 8, 2020, BCTCS, Swansea, United Kingdom

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Outline

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

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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.

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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}.

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Fixed-Parameter Tractability (FPT)

A

(x, k) Yes if (x, k) 2 L

No, otherwise

  • Algorithm A runs in f(k) · |x|c time.
  • A is called Fixed Parameter Algorithm.

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Hardness in Parameterized Complexity

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Hardness in Parameterized Complexity

FPT ⊆ W[1] ⊆ W[2] ⊆ . . . ⊆ XP.

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Kernelization

(x, k) (x0, k0)

  • Preprocessing takes poly(|x|, k) time.
  • (x, k) 2 L if and only if (x0, k0) 2 L.
  • |x0| + k0 ≤ g(k).
  • If g(k) = poly(k), then we say that L has a polynomial kernel.

Preprocess

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Graph Coloring

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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)?

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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.

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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.

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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.

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Precoloring Extension

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

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Precoloring Extension

Pre-coloring Extension Input: A graph G, and a precoloring λP : X → Q for X ⊆ V (G) where Q is a set of colors. Goal: Can λP 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.

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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) →

  • u∈V (G)

L(u) such that for every u ∈ V (G), λ(u) ∈ L(u)?

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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) →

  • u∈V (G)

L(u) such that for every u ∈ V (G), λ(u) ∈ L(u)?

  • List Coloring is polynomial time solvable in clique, and

cluster graph.

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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) →

  • u∈V (G)

L(u) such 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.

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Outline

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

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Our problems and results

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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 λP : X → Q for X ⊆ V (G) where Q is a set of colors. Parameter: k Question: Can λP be extended to a proper coloring of G using colors from only Q?

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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 λP : X → Q for X ⊆ V (G) where Q is a set of colors. Parameter: k Question: Can λP 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?

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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

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Pre-coloring Extension

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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 λP : X → Q for X ⊆ V (G) where Q is a set of colors. Parameter: k Question: Can λP be extended to a proper coloring of G using colors from only Q?

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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 λP : X → Q for X ⊆ V (G) where Q is a set of colors. Parameter: k Question: Can λP 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.

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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 λP : X → Q for X ⊆ V (G) where Q is a set of colors. Parameter: k Question: Can λP 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.

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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 λP : X → Q for X ⊆ V (G) where Q is a set of colors. Parameter: k Question: Can λP 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 3k vertices.

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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?

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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?

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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′)

  • f (n − k)-Regular List Coloring such that k′ = 2k.

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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?

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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.

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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.

  • We prove that List Coloring Clique Modulator

admits a randomized algorithm running in time O∗(2k).

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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.

  • We prove that List Coloring Clique Modulator

admits a randomized algorithm running in time O∗(2k).

  • We prove that List Coloring Clique Modulator

admits no polynomial kernel unless NP ⊆ coNP/poly.

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List Coloring

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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?

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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?

  • Banik, Jacob, Paliwal, and Raman [IWOCA 2019] proved

that (n − k)-Regular List Coloring is FPT and asked if (n − k)-Regular List Coloring admits a polynomial kernel.

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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?

  • Banik, Jacob, Paliwal, and Raman [IWOCA 2019] proved

that (n − k)-Regular List Coloring is FPT and asked if (n − k)-Regular List Coloring admits a polynomial kernel.

  • We answer positively by proving that (n − k)-Regular

List Coloring admits a polynomial kernel with O(k2) vertices and colors.

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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?

  • Banik, Jacob, Paliwal, and Raman [IWOCA 2019] proved

that (n − k)-Regular List Coloring is FPT and asked if (n − k)-Regular List Coloring admits a polynomial kernel.

  • We answer positively by proving that (n − k)-Regular

List Coloring admits a polynomial kernel with O(k2) vertices and colors.

  • We also provide a compression to a variation of the

problem with 11k vertices and O(k2) colors, encodable in O(k2 log2 k) bits.

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Outline

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

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Precoloring Extension

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Precoloring Extension

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

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Precoloring Extension

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Precoloring Extension

  • Rule 1: If a vertex v ∈ D′ has less than |Q| neighbors in

G, then delete v from G.

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Precoloring Extension

  • Rule 1: If a vertex v ∈ D′ has less than |Q| neighbors in

G, then delete v from G.

  • When Rule 1 is not applicable, and |Q| ≥ |C|, hence

|CN| ≤ k2.

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Precoloring Extension

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Precoloring Extension

  • Construct auxiliary bipartite graph J = (CN, D).

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Precoloring Extension

  • Construct auxiliary bipartite graph J = (CN, D).
  • Rule 2: Let A ⊆ CN be an inclusion-wise minimal set such

that |A| > NJ(A). Remove D′ ∩ NJ(A) from G.

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Precoloring Extension

  • Construct auxiliary bipartite graph J = (CN, D).
  • Rule 2: Let A ⊆ CN be an inclusion-wise minimal set such

that |A| > NJ(A). Remove D′ ∩ NJ(A) from G.

  • If Rule 2 is not applicable, then |CN| ≤ |D| ≤ k.

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Precoloring Extension

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Precoloring Extension

  • Rule 3: Let v ∈ C be a pre-colored vertex with color

λP (v). Then remove the vertex set λ−1

P (λP (v)) from G and

remove λP (v) from Q.

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Precoloring Extension

  • Rule 3: Let v ∈ C be a pre-colored vertex with color

λP (v). Then remove the vertex set λ−1

P (λP (v)) from G and

remove λP (v) from Q.

  • When Rule 3 is not applicable, there is no pre-colored

vertex in C′.

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Precoloring Extension

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Precoloring Extension

  • Construct bipartite graph H = (C, P) as above.

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Precoloring Extension

  • Construct bipartite graph H = (C, P) as above.
  • Rule 4: Remove the vertices of C′ \ CM and arbitrary

|C′ \ CM| colors from Q.

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Precoloring Extension

  • Construct bipartite graph H = (C, P) as above.
  • Rule 4: Remove the vertices of C′ \ CM and arbitrary

|C′ \ CM| colors from Q.

  • If Rule 4 is not applicable, then |C′| ≤ |P| ≤ |DP | ≤ k.

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Precoloring Extension: Result

DP

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D0

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C0

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CN

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DP : precolored vertices from D.

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CN = {v 2 C|9u 2 D0, uv / 2 E(G)}

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23

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SLIDE 59

Precoloring Extension: Result

DP

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D0

<latexit sha1_base64="o3+5FgCYSoFSC79UwMBzCzdx5Nk=">AB6XicbVBNS8NAEJ34WetX1aOXxSJ6KokU9FjUg8cq9gPaUDbTbt0swm7E6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjVjDdYLGPdDqjhUijeQIGStxPNaRI3gpGN1O/9cS1EbF6xHC/YgOlAgFo2ilh9uzXqnsVtwZyDLxclKGHPVe6avbj1kacYVMUmM6npugn1GNgk+KXZTwxPKRnTAO5YqGnHjZ7NLJ+TUKn0SxtqWQjJTf09kNDJmHAW2M6I4NIveVPzP6QYXvmZUEmKXLH5ojCVBGMyfZv0heYM5dgSyrSwtxI2pJoytOEUbQje4svLpHlR8aqV6n21XLvO4yjAMZzAOXhwCTW4gzo0gEIz/AKb87IeXHenY9564qTzxzBHzifP/nzjQA=</latexit>

C0

<latexit sha1_base64="2VjUfjEmMuL2TlAnAxqiOcAp6k=">AB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMdiLx6rWFtoQ9lsJ+3SzSbsboQS+g+8eFDEq/Im/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fDoUcepYthisYhVJ6AaBZfYMtwI7CQKaRQIbAfjxsxvP6HSPJYPZpKgH9Gh5CFn1FjpvnHeL1fcqjsHWSVeTiqQo9kvf/UGMUsjlIYJqnXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/m186JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZIalGyxKEwFMTGZvU0GXCEzYmIJZYrbWwkbUWZseGUbAje8sur5PGy6tWqtbtapX6Tx1GEziFC/DgCupwC01oAYMQnuEV3pyx8+K8Ox+L1oKTzxzDHzifP/hujP8=</latexit>

CN

<latexit sha1_base64="8Y8jOiw5euslU/ApJU4cPLQ+m8g=">AB6nicbVBNS8NAEJ34WetX1aOXxSJ4KokU9FjsxZNUtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq7aytb2xubRd2irt7+weHpaPjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7G9ZnfkKleSwfzSRBP6JDyUPOqLHSQ71/1y+V3Yo7B1klXk7KkKPRL31BjFLI5SGCap13MT42dUGc4ETou9VGNC2ZgOsWupBFqP5ufOiXnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhNd+xmWSGpRsShMBTExmf1NBlwhM2JiCWK21sJG1FmbHpFG0I3vLq6R1WfGqlep9tVy7yeMowCmcwQV4cAU1uIUGNIHBEJ7hFd4c4bw4787HonXNyWdO4A+czx/pHY2P</latexit>

DP : precolored vertices from D.

<latexit sha1_base64="szfJv8naVZCoU6C5tDeu4dVFZHE=">ACDXicbVBLS8NAGNzUV62vqEcvi43gKSloIiHgj14rGAf0Iaw2WzapZtNyG6EvoHvPhXvHhQxKt3b/4bN2kO2jqwMx8rx0vZlRIy/rWKmvrG5tb1e3azu7e/oF+eNQTUZpg0sURi5KBhwRhlJOupJKRQZwQFHqM9L3pTe73H0giaMTv5SwmTojGnAYUI6kVzeMtsxrqBqKoYRH6pySTERMEiEBptw3T1umVaBeAqsUtSByU6rv418iOchoRLzJAQ9uKpZOhfDAj89oFSRGeIrGZKgoRyERTlYcMIdnSvFhECXqcQkL9XdHhkIhZqGnKkMkJ2LZy8X/vGEqg0snozxOJeF4sShIGZQRzKOBPlUhSDZTBOGEqlshnqAEYakCrKkQ7OUvr5Jew7SbZvOuUW9dl3FUwQk4BefABhegBW5B3QBo/gGbyCN+1Je9HetY9FaUre47BH2ifP7GAmgk=</latexit>

CN = {v 2 C|9u 2 D0, uv / 2 E(G)}

<latexit sha1_base64="rotxQ+zulxF8uyUI+DiAJkQRvhk=">ACHicbVDLSgMxFM3UV62vUZdugkWsIGVGC7pQKFTRlVSwD+iUknTNjSTGZJMsYz9EDf+ihsXirhxIfg3ptNZaOuBwLn3MvNPW7AqFSW9W2k5uYXFpfSy5mV1bX1DXNzqyr9UGBSwT7zRd1FkjDKSUVRxUg9EAR5LiM1t18a+7UBEZL6/E4NA9L0UJfTDsVIalHpdaN/AcOtEAOpTD0gN0yL1eK2EYCxf7hzDUHveVri5zVwfOqGVmrbwVA84SOyFZkKDcMj+dto9Dj3CFGZKyYVuBakZIKIoZGWcUJIA4T7qkoamHlENqP4uBHc0obdnyhH1cwVn9PRMiTcui5utNDqienvbH4n9cIVe0GVEehIpwPFnUCRlUPhwnBdtUEKzYUBOEBdV/hbiHBMJK5nRIdjTJ8+S6lHeLuQLt4Vs8SyJIw12wC7IARucgCK4BmVQARg8gmfwCt6MJ+PFeDc+Jq0pI5nZBn9gfP0AjOe1g=</latexit>
  • |CN| ≤ |D| ≤ k, |C′| ≤ |DP | ≤ k.

23

slide-60
SLIDE 60

Precoloring Extension: Result

DP

<latexit sha1_base64="BIdkY30ZbdGeIKrpgwq/cENTF1I=">AB6nicbVDLSgMxFL1TX7W+qi7dBIvgqsxIQZdFXbisaB/QDiWTZtrQTGZI7ghl6Ce4caGIW7/InX9j2s5CWw8EDuecS+49QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3k0p1EgeTsY38z89hPXRsTqEScJ9yM6VCIUjKVHm7jX654lbdOcgq8XJSgRw2/9UbxCyNuEImqTFdz03Qz6hGwSflnqp4QlYzrkXUsVjbjxs/mqU3JmlQEJY2fQjJXf09kNDJmEgU2GVEcmWVvJv7ndVMr/xMqCRFrtjiozCVBGMyu5sMhOYM5cQSyrSwuxI2opoytO2UbAne8smrpHVR9WrV2n2tUr/O6yjCZzCOXhwCXW4gwY0gcEQnuEV3hzpvDjvzsciWnDymWP4A+fzB+2rjZI=</latexit>

D0

<latexit sha1_base64="o3+5FgCYSoFSC79UwMBzCzdx5Nk=">AB6XicbVBNS8NAEJ34WetX1aOXxSJ6KokU9FjUg8cq9gPaUDbTbt0swm7E6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjVjDdYLGPdDqjhUijeQIGStxPNaRI3gpGN1O/9cS1EbF6xHC/YgOlAgFo2ilh9uzXqnsVtwZyDLxclKGHPVe6avbj1kacYVMUmM6npugn1GNgk+KXZTwxPKRnTAO5YqGnHjZ7NLJ+TUKn0SxtqWQjJTf09kNDJmHAW2M6I4NIveVPzP6QYXvmZUEmKXLH5ojCVBGMyfZv0heYM5dgSyrSwtxI2pJoytOEUbQje4svLpHlR8aqV6n21XLvO4yjAMZzAOXhwCTW4gzo0gEIz/AKb87IeXHenY9564qTzxzBHzifP/nzjQA=</latexit>

C0

<latexit sha1_base64="2VjUfjEmMuL2TlAnAxqiOcAp6k=">AB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMdiLx6rWFtoQ9lsJ+3SzSbsboQS+g+8eFDEq/Im/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fDoUcepYthisYhVJ6AaBZfYMtwI7CQKaRQIbAfjxsxvP6HSPJYPZpKgH9Gh5CFn1FjpvnHeL1fcqjsHWSVeTiqQo9kvf/UGMUsjlIYJqnXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/m186JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZIalGyxKEwFMTGZvU0GXCEzYmIJZYrbWwkbUWZseGUbAje8sur5PGy6tWqtbtapX6Tx1GEziFC/DgCupwC01oAYMQnuEV3pyx8+K8Ox+L1oKTzxzDHzifP/hujP8=</latexit>

CN

<latexit sha1_base64="8Y8jOiw5euslU/ApJU4cPLQ+m8g=">AB6nicbVBNS8NAEJ34WetX1aOXxSJ4KokU9FjsxZNUtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq7aytb2xubRd2irt7+weHpaPjlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7G9ZnfkKleSwfzSRBP6JDyUPOqLHSQ71/1y+V3Yo7B1klXk7KkKPRL31BjFLI5SGCap13MT42dUGc4ETou9VGNC2ZgOsWupBFqP5ufOiXnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhNd+xmWSGpRsShMBTExmf1NBlwhM2JiCWK21sJG1FmbHpFG0I3vLq6R1WfGqlep9tVy7yeMowCmcwQV4cAU1uIUGNIHBEJ7hFd4c4bw4787HonXNyWdO4A+czx/pHY2P</latexit>

DP : precolored vertices from D.

<latexit sha1_base64="szfJv8naVZCoU6C5tDeu4dVFZHE=">ACDXicbVBLS8NAGNzUV62vqEcvi43gKSloIiHgj14rGAf0Iaw2WzapZtNyG6EvoHvPhXvHhQxKt3b/4bN2kO2jqwMx8rx0vZlRIy/rWKmvrG5tb1e3azu7e/oF+eNQTUZpg0sURi5KBhwRhlJOupJKRQZwQFHqM9L3pTe73H0giaMTv5SwmTojGnAYUI6kVzeMtsxrqBqKoYRH6pySTERMEiEBptw3T1umVaBeAqsUtSByU6rv418iOchoRLzJAQ9uKpZOhfDAj89oFSRGeIrGZKgoRyERTlYcMIdnSvFhECXqcQkL9XdHhkIhZqGnKkMkJ2LZy8X/vGEqg0snozxOJeF4sShIGZQRzKOBPlUhSDZTBOGEqlshnqAEYakCrKkQ7OUvr5Jew7SbZvOuUW9dl3FUwQk4BefABhegBW5B3QBo/gGbyCN+1Je9HetY9FaUre47BH2ifP7GAmgk=</latexit>

CN = {v 2 C|9u 2 D0, uv / 2 E(G)}

<latexit sha1_base64="rotxQ+zulxF8uyUI+DiAJkQRvhk=">ACHicbVDLSgMxFM3UV62vUZdugkWsIGVGC7pQKFTRlVSwD+iUknTNjSTGZJMsYz9EDf+ihsXirhxIfg3ptNZaOuBwLn3MvNPW7AqFSW9W2k5uYXFpfSy5mV1bX1DXNzqyr9UGBSwT7zRd1FkjDKSUVRxUg9EAR5LiM1t18a+7UBEZL6/E4NA9L0UJfTDsVIalHpdaN/AcOtEAOpTD0gN0yL1eK2EYCxf7hzDUHveVri5zVwfOqGVmrbwVA84SOyFZkKDcMj+dto9Dj3CFGZKyYVuBakZIKIoZGWcUJIA4T7qkoamHlENqP4uBHc0obdnyhH1cwVn9PRMiTcui5utNDqienvbH4n9cIVe0GVEehIpwPFnUCRlUPhwnBdtUEKzYUBOEBdV/hbiHBMJK5nRIdjTJ8+S6lHeLuQLt4Vs8SyJIw12wC7IARucgCK4BmVQARg8gmfwCt6MJ+PFeDc+Jq0pI5nZBn9gfP0AjOe1g=</latexit>
  • |CN| ≤ |D| ≤ k, |C′| ≤ |DP | ≤ k.
  • Hence, Pre-Coloring Extension Clique Modulator

has a kernel with 3k vertices.

23

slide-61
SLIDE 61

Outline

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

24

slide-62
SLIDE 62

Conclusions

25

slide-63
SLIDE 63

Conclusions

  • Can we derandomize our algorithms for List Coloring

Clique Modulator?

25

slide-64
SLIDE 64

Conclusions

  • Can we derandomize our algorithms for List Coloring

Clique Modulator?

  • For (n − k)-Regular List Coloring, can we get a kernel

with O(k) veritces?

25

slide-65
SLIDE 65

THANK YOU

26