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On-Line Graph Coloring On-line coloring Competitiveness Results Example On-line Graph Coloring Iwona Cieslik´ Algorithmics Research Group, Jagiellonian University, Cracow, Poland Chambéry, June 2005 Iwona Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example On-line Graph Coloring Example Iwona Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example On-line Graph Coloring Example Spoiler puts first vertex. Algorithm colors it by red. Iwona Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example On-line cliques covering Example Iwona Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example On-line cliques covering Example Spoiler puts first vertex. Algorithm covers it by red. Iwona Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example Forcing subgraphs There are some families of graphs for which there exists no on-line algorithm with any competitive function. Iwona Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example Forcing subgraphs There are some families of graphs for which there exists no on-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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example Forcing subgraphs There are some families of graphs for which there exists no on-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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example Examples b b b b b b b b b b b b b b b b b b b b K1,3 K1,3 − free K2,4 K2,4 − free Iwona Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example The table of competitiveness Graph Method Coloring Cliques of giving covering K ∞ s s-free - 2 s ≥ 3 Iwona Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example The table of competitiveness Graph Method Coloring Cliques of giving covering ∞ 3 Bipartite - 2 [Gyárfás Lehel] 3 Bipartite connected 1 2 K ∞ s s-free - 2 s ≥ 3 Iwona Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example The table of competitiveness Graph Method Coloring Cliques of giving covering ∞ 3 Bipartite - 2 [Gyárfás Lehel] 3 Bipartite connected 1 2 K ∞ s s-free - 2 s ≥ 3 K ∞ s s-free connected 2 s ≥ 3 Iwona Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring 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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example ON-LINE COLORING for C4-free graphs Theorem (Cieslik)´ 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 Cieslik´ On-line Graph Coloring On-line coloring Competitiveness Results Example ON-LINE COLORING for C4-free graphs Theorem (Cieslik)´ The competitive function of the best on-line cliques covering algorithm for C4-free graphs is at least: O( opt(G) · lg opt(G)).
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