On-line Graph Coloring Iwona Cie slik Algorithmics Research Group, - - PowerPoint PPT Presentation

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On-line Graph Coloring Iwona Cie slik Algorithmics Research Group, - - PowerPoint PPT Presentation

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On-line coloring Competitiveness Results Example On-line Graph Coloring Iwona Cie slik Algorithmics Research Group, Jagiellonian University, Cracow, Poland Chambry, June 2005 Iwona Cie slik On-line Graph Coloring On-line coloring

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On-line coloring Competitiveness Results Example

On-line Graph Coloring

Iwona Cie´ slik

Algorithmics Research Group, Jagiellonian University, Cracow, Poland

Chambéry, June 2005

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Graph Coloring

Definition A vertex coloring of a graph is a function assigns to every vertex a color in such a way that the adjacency vertices have different colors.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Graph Coloring

Definition A vertex coloring of a graph is a function assigns to every vertex a color in such a way that the adjacency vertices have different colors. Example

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line coloring like a game

an infinite board 2 players: Spoiler and Algorithm Spoiler gives a graph. At each step he draws a single, new vertex. Algorithm assigns colors to vertices. Algorithm wants to use as few colors as possible. Spoiler wants to cheat Algorithm, and force him to use as many colors as possible.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line Graph Coloring

Example

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line Graph Coloring

Example Spoiler puts first vertex. Algorithm colors it by red.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line Graph Coloring

Example Spoiler puts first vertex. Algorithm colors it by red. Spoiler puts second vertex. Algorithm colors it by blue.

Iwona Cie´ slik On-line Graph Coloring

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SLIDE 8

On-line coloring Competitiveness Results Example

On-line Graph Coloring

Example Spoiler puts first vertex. Algorithm colors it by red. Spoiler puts second vertex. Algorithm colors it by blue. Spoiler puts third vertex. Algorithm colors it by red.

Iwona Cie´ slik On-line Graph Coloring

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SLIDE 9

On-line coloring Competitiveness Results Example

On-line Graph Coloring

Example Spoiler puts first vertex. Algorithm colors it by red. Spoiler puts second vertex. Algorithm colors it by blue. Spoiler puts third vertex. Algorithm colors it by red. Spoiler puts fourth vertex. Algorithm colors it by yellow.

Iwona Cie´ slik On-line Graph Coloring

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SLIDE 10

On-line coloring Competitiveness Results Example

On-line Graph Coloring

Example Spoiler puts first vertex. Algorithm colors it by red. Spoiler puts second vertex. Algorithm colors it by blue. Spoiler puts third vertex. Algorithm colors it by red. Spoiler puts fourth vertex. Algorithm colors it by yellow. Spoiler puts fifth vertex. Algorithm colors it by green.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Minimal cliques covering

Similarly, we can define an on-line cliques covering problem. Definition Minimal cliques covering problem asks for finding a partition of vertices of a graph into a minimal number of cliques.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Minimal cliques covering

Similarly, we can define an on-line cliques covering problem. Definition Minimal cliques covering problem asks for finding a partition of vertices of a graph into a minimal number of cliques. Example

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line cliques covering

Example

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line cliques covering

Example Spoiler puts first vertex. Algorithm covers it by red.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line cliques covering

Example Spoiler puts first vertex. Algorithm covers it by red. Spoiler puts second vertex. Algorithm covers it by red.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line cliques covering

Example Spoiler puts first vertex. Algorithm covers it by red. Spoiler puts second vertex. Algorithm covers it by red. Spoiler puts third vertex. Algorithm covers it by blue.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line cliques covering

Example Spoiler puts first vertex. Algorithm covers it by red. Spoiler puts second vertex. Algorithm covers it by red. Spoiler puts third vertex. Algorithm covers it by blue. Spoiler puts fourth vertex. Algorithm covers it by blue.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

On-line cliques covering

Example Spoiler puts first vertex. Algorithm covers it by red. Spoiler puts second vertex. Algorithm covers it by red. Spoiler puts third vertex. Algorithm covers it by blue. Spoiler puts fourth vertex. Algorithm covers it by blue. Spoiler puts fifth vertex. Algorithm covers it by green.

Iwona Cie´ slik On-line Graph Coloring

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SLIDE 19

On-line coloring Competitiveness Results Example

On-line cliques covering

Example Spoiler puts first vertex. Algorithm covers it by red. Spoiler puts second vertex. Algorithm covers it by red. Spoiler puts third vertex. Algorithm covers it by blue. Spoiler puts fourth vertex. Algorithm covers it by blue. Spoiler puts fifth vertex. Algorithm covers it by green. Spoiler puts sixth vertex. Algorithm covers it by yellow.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Competitiveness

Definition An on-line algorithm A is competitive for the family F if there is a function f such that for every G ∈ F A(G) ≤ f(opt(G))

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Competitiveness

Definition An on-line algorithm A is competitive for the family F if there is a function f such that for every G ∈ F A(G) ≤ f(opt(G)) Definition An on-line algorithm A is c-competitive for the family F if there exist a and c such that for every G ∈ F A(G) ≤ c · opt(G) + a We say that c is a competitive ratio.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

How can we show that on-line problem has a competitive ratio equal to c?

Find an on-line c-competitive algorithm A, Show that there’s no on-line algorithm which is better. That is, indicate a proper forcing strategy for Spoiler. Similarly, we can show that the on-line problem has more complex competitive function.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Forcing subgraphs

There are some families of graphs for which there exists no

  • n-line algorithm with any competitive function.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Forcing subgraphs

There are some families of graphs for which there exists no

  • n-line algorithm with any competitive function.

There are some families of graphs for which Spoiler is able to force only a few additional colors.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Forcing subgraphs

There are some families of graphs for which there exists no

  • n-line algorithm with any competitive function.

There are some families of graphs for which Spoiler is able to force only a few additional colors. In order that Spoiler can force a large number of colors, the given graph should contain the forcing subgraphs.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

H-free graphs

Definition G - H-free graph

  • it does not contain any induced subgraph isomorphic to H

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

H-free graphs

Definition G - H-free graph

  • it does not contain any induced subgraph isomorphic to H

Examples Ks-free graphs, Ks,t-free graphs ... Ks - a complete graph (clique) with s vertices Ks,t - a complete bipartite graph (X, Y, E) : ♯X = s, ♯Y = t

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Examples K1,3 K1,3 − free K2,4 K2,4 − free

b b b b b b b b b b b b b b b b b b b b Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

The table of competitiveness

Graph Method Coloring Cliques

  • f giving

covering Ks-free

  • ∞

s 2

s ≥ 3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

The table of competitiveness

Graph Method Coloring Cliques

  • f giving

covering Bipartite

  • ∞

3 2

[Gyárfás Lehel] Bipartite connected 1

3 2

Ks-free

  • ∞

s 2

s ≥ 3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

The table of competitiveness

Graph Method Coloring Cliques

  • f giving

covering Bipartite

  • ∞

3 2

[Gyárfás Lehel] Bipartite connected 1

3 2

Ks-free

  • ∞

s 2

s ≥ 3 Ks-free connected ∞

s 2

s ≥ 3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Graph Method of giving Coloring Cliques covering K1,t-free

  • t − 1

∞ t ≥ 3 K1,t-free connected t − 1 ∞ t ≥ 3 Ks,t-free

  • ∞

∞ s ≥ 2, t ≥ 3 Ks,t-free connected ∞ ∞ s ≥ 2, t ≥ 3 K2,2-free

  • ∞

??? = C4-free

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

ON-LINE COLORING for C4-free graphs

Theorem (Gyárfás, Lehel, 1988) An upper bound for a competitive function of an on-line coloring for C4-free graphs is at most

2opt(G) − 1.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

ON-LINE COLORING for C4-free graphs

Theorem (Gyárfás, Lehel, 1988) An upper bound for a competitive function of an on-line coloring for C4-free graphs is at most

2opt(G) − 1.

Question What about a lower bound ??

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

ON-LINE COLORING for C4-free graphs

Theorem (Cie´ slik) The competitive function of the best on-line cliques covering algorithm for C4-free graphs is at least:

O( opt(G) · lg opt(G) ).

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

ON-LINE COLORING for C4-free graphs

Theorem (Cie´ slik) The competitive function of the best on-line cliques covering algorithm for C4-free graphs is at least:

O( opt(G) · lg opt(G) ).

Theorem (Cie´ slik) The competitive function of the best greedy on-line cliques covering algorithm for C4-free graphs is at least quadratic:

  • pt(G) + 1

2.

  • Iwona Cie´

slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Example

Theorem (Cie´ slik) For every on-line coloring algorithm A there exists a graph G such that A(G) ≥ (t − 1) · opt(G) − t + 2. As an example we show a construction of the 4-colorable K1,4-free graph G for which the on-line coloring greedy algorithm with the First Fit strategy uses 3 · opt(G) − 2 = 3 · 4 − 2 = 10 colors.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

A1 A2 A3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b

b b

v1 A1 A2 A3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b

b b

v1 v2 A1 A2 A3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

v1 v2 v3 A1 A2 A3

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

v1 v2 v3 C1 B1

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

C1 B1 B2 B3 B4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b

b b b b

C1 C2 B1 B2 B3 B4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b

b b b b

b b b

C1 C2 C3 B1 B2 B3 B4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

b b

b b

b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b

b b b b

C1 C2 C3 C4 B1 B2 B3 B4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b

b

C1 C2 C3 C4 B1 B2 B3 B4 D1 D2 D3 D4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b

b b

7 C1 C2 C3 C4 B1 B2 B3 B4 D1 D2 D3 D4

Iwona Cie´ slik On-line Graph Coloring

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SLIDE 51

On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b

b

7

b b

8 C1 C2 C3 C4 B1 B2 B3 B4 D1 D2 D3 D4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b

b b

b

7

b b

8

b 9 b b

C1 C2 C3 C4 B1 B2 B3 B4 D1 D2 D3 D4

Iwona Cie´ slik On-line Graph Coloring

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SLIDE 53

On-line coloring Competitiveness Results Example b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b

b

7

b b

8

b b 9

10 9 10 C1 C2 C3 C4 B1 B2 B3 B4 D1 D2 D3 D4

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Theorem (Cie´ slik) For every on-line coloring algorithm A there exists a graph G given in a connected way such that A(G) ≥ (t − 1) · opt(G) − t + 2.

Iwona Cie´ slik On-line Graph Coloring

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On-line coloring Competitiveness Results Example

Thank you for your attention.

Iwona Cie´ slik On-line Graph Coloring