Finer Tight Bounds for Coloring on Clique-width Michael Lampis LAMSADE Universit´ e Paris Dauphine ICALP 2018
Coloring Input: Graph G = ( V, E ) n vertices k colors Question: Can we partition V into k independent sets? Parameterized Approximation Schemes 2 / 18
Coloring Input: Graph G = ( V, E ) n vertices k colors Question: Can we partition V into k independent sets? Parameterized Approximation Schemes 2 / 18
Coloring Input: Graph G = ( V, E ) n vertices k colors Question: Can we partition V into k independent sets? Note: For the rest of this talk, k denotes the number of colors . Problem NP-hard for any k ≥ 3 : We look at graphs with restricted structure. Parameterized Approximation Schemes 2 / 18
Finer Tight Bounds? • What is a “finer” tight bound? Parameterized Approximation Schemes 3 / 18
Finer Tight Bounds? • Tight bound: complexity-theoretic bound that “matches” running time of existing algorithm. • Finer bounds: • Increased “granularity”. • More precise about secondary parameters. Parameterized Approximation Schemes 3 / 18
Finer Tight Bounds? • Tight bound: complexity-theoretic bound that “matches” running time of existing algorithm. • Finer bounds: • Increased “granularity”. • More precise about secondary parameters. Coloring • We know the “correct” complexity of Coloring for clique-width . . . ≈ k 2 w (more details in a bit) • • This bound is only tight for k sufficiently large. • What is the exact complexity of 3 -coloring, 4 -coloring for clique-width? Parameterized Approximation Schemes 3 / 18
Finer Tight Bounds? • Tight bound: complexity-theoretic bound that “matches” running time of existing algorithm. • Finer bounds: • Increased “granularity”. • More precise about secondary parameters. Coloring • We know the “correct” complexity of Coloring for clique-width . . . ≈ k 2 w (more details in a bit) • • This bound is only tight for k sufficiently large. • What is the exact complexity of 3 -coloring, 4 -coloring for clique-width? In this talk we show that, under the SETH, the correct complexity of k -Coloring for clique-width is Parameterized Approximation Schemes 3 / 18
Finer Tight Bounds? • Tight bound: complexity-theoretic bound that “matches” running time of existing algorithm. • Finer bounds: • Increased “granularity”. • More precise about secondary parameters. Coloring • We know the “correct” complexity of Coloring for clique-width . . . ≈ k 2 w (more details in a bit) • • This bound is only tight for k sufficiently large. • What is the exact complexity of 3 -coloring, 4 -coloring for clique-width? In this talk we show that, under the SETH, the correct complexity of k -Coloring for clique-width is Parameterized Approximation Schemes 3 / 18
Finer Tight Bounds? • Tight bound: complexity-theoretic bound that “matches” running time of existing algorithm. • Finer bounds: • Increased “granularity”. • More precise about secondary parameters. Coloring • We know the “correct” complexity of Coloring for clique-width . . . ≈ k 2 w (more details in a bit) • • This bound is only tight for k sufficiently large. • What is the exact complexity of 3 -coloring, 4 -coloring for clique-width? In this talk we show that, under the SETH, the correct complexity of k -Coloring for clique-width is c w k . Parameterized Approximation Schemes 3 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path • w extra vertices, arbitrarily connected to each other Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path • w extra vertices, arbitrarily connected to each other • and arbitrary edges between these two parts Interesting case: w << n . Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path 3-Coloring algorithm on these graphs: • Guess a valid coloring of the w non-path vertices • Try to extend it to a coloring of the whole graph (easy!) Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path 3-Coloring algorithm on these graphs: • Guess a valid coloring of the w non-path vertices • Try to extend it to a coloring of the whole graph (easy!) Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path 3-Coloring algorithm on these graphs: • Guess a valid coloring of the w non-path vertices • Try to extend it to a coloring of the whole graph (easy!) Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path 3-Coloring algorithm on these graphs: • Guess a valid coloring of the w non-path vertices • Try to extend it to a coloring of the whole graph (easy!) Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth Consider this ( very very special ) class of graphs of treewidth w : • The graph consists of a long path 3-Coloring algorithm on these graphs: • Guess a valid coloring of the w non-path vertices • Try to extend it to a coloring of the whole graph (easy!) • Either found a valid coloring, or try another coloring for w vertices. Running time: 3 w Parameterized Approximation Schemes 4 / 18
The story so far: Treewidth • Graphs of treewidth w are much more general than the graphs of the previous slide. • Algorithm generalizes easily (DP) Running time: k w . • Parameterized Approximation Schemes 5 / 18
The story so far: Treewidth • Graphs of treewidth w are much more general than the graphs of the previous slide. • Algorithm generalizes easily (DP) Running time: k w . • Can we do better? Parameterized Approximation Schemes 5 / 18
The story so far: Treewidth • Graphs of treewidth w are much more general than the graphs of the previous slide. • Algorithm generalizes easily (DP) Running time: k w . • Can we do better? Parameterized Approximation Schemes 5 / 18
The story so far: Treewidth • Graphs of treewidth w are much more general than the graphs of the previous slide. • Algorithm generalizes easily (DP) Running time: k w . • Can we do better? Previous Work: • Lokshtanov, Marx, Saurabh, SODA’11 • Jaffke and Jansen, CIAC ’17 Result: (SETH) → cannot do ( k − ǫ ) w , for any k, ǫ , even for Paths+ w ! Very fine , completely tight bound! Note: SETH ≈ SAT has no 1 . 999 n algorithm. Parameterized Approximation Schemes 5 / 18
The story so far: Treewidth • Graphs of treewidth w are much more general than the graphs of the previous slide. • Algorithm generalizes easily (DP) Running time: k w . • Can we do better? Previous Work: • Lokshtanov, Marx, Saurabh, SODA’11 • Jaffke and Jansen, CIAC ’17 Result: (SETH) → cannot do ( k − ǫ ) w , for any k, ǫ , even for Paths+ w ! Very fine , completely tight bound! Note: SETH ≈ SAT has no 1 . 999 n algorithm. Parameterized Approximation Schemes 5 / 18
The story so far: Clique-width • Clique-width is the second most widely studied graph width. • Intuition: Treewidth + Some dense graphs. • Definition in next slide. Summary of what is known for k -Coloring on graphs of clique-width w : Algorithm in k 2 O ( w ) (Kobler and Rotics DAM ’03) • Algorithm in 4 k · w (Kobler and Rotics DAM ’03) • • W-hard parameterized by w (Fomin, Golovach, Lokshtanov, and Saurabh SICOMP ’10) ETH LB of n 2 o ( w ) (Golovach, Lokshtanov, Saurabh, Zehavi SODA’18) • Parameterized Approximation Schemes 6 / 18
The story so far: Clique-width • Clique-width is the second most widely studied graph width. • Intuition: Treewidth + Some dense graphs. • Definition in next slide. Summary of what is known for k -Coloring on graphs of clique-width w : Algorithm in k 2 O ( w ) (Kobler and Rotics DAM ’03) • Algorithm in 4 k · w (Kobler and Rotics DAM ’03) • • W-hard parameterized by w (Fomin, Golovach, Lokshtanov, and Saurabh SICOMP ’10) ETH LB of n 2 o ( w ) (Golovach, Lokshtanov, Saurabh, Zehavi SODA’18) • Remark: Last LB is tight (!), but requires k to be large (otherwise contradicts second algorithm) Story not as clear as treewidth (yet). . . Parameterized Approximation Schemes 6 / 18
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