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Bidimensionality Theory and Applications Bidimensionality theory and applications Alexandros Angelopoulos M.P.L.A. July 1, 2014 Outline The idea Subexponential parametrized algorithms PTAS / EPTAS Kernelization Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 2/23 Flashback: separators and treewidth S R Q Figure: Scheme for DP applied on graphs 4 Usually try to bound the size of Sp. If, for instance, we have the case of the treewidth of planar graphs (O( n)), we can design FPT algorithms with sub-exponential parameterized dependence. Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 3/23 Flashback: separators and treewidth S R Q Figure: Scheme for DP applied on graphs 4 Usually try to bound the size of Sp. If, for instance, we have the case of the treewidth of planar graphs (O( n)), we can design FPT algorithms with sub-exponential parameterized dependence. Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 3/23 Flashback: we ª treewidth 4 tw is often sublinear in the solution size k (of planar graphs) 4 It supports a \win-win" approach: u check (roughly) if tw(G) > o(k) and if true output \NO" u else, use DP and design something that runs in 2o(k)n steps. Remember: there exists an algorithm that, with input a graph G and an integer c either reports that tw(G) > c or outputs a tree decomposition of G of width at most 5c in 2O(c)n steps, as a result based on Bodlaender, [1]. But: 8 To get that tw(G) = o(k) for a sole parameter k and graph class G, we need to use our best combinatorics... Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 4/23 Flashback: we ª treewidth 4 tw is often sublinear in the solution size k (of planar graphs) 4 It supports a \win-win" approach: u check (roughly) if tw(G) > o(k) and if true output \NO" u else, use DP and design something that runs in 2o(k)n steps. Remember: there exists an algorithm that, with input a graph G and an integer c either reports that tw(G) > c or outputs a tree decomposition of G of width at most 5c in 2O(c)n steps, as a result based on Bodlaender, [1]. But: 8 To get that tw(G) = o(k) for a sole parameter k and graph class G, we need to use our best combinatorics... Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 4/23 Flashback: we ª treewidth 4 tw is often sublinear in the solution size k (of planar graphs) 4 It supports a \win-win" approach: u check (roughly) if tw(G) > o(k) and if true output \NO" u else, use DP and design something that runs in 2o(k)n steps. Remember: there exists an algorithm that, with input a graph G and an integer c either reports that tw(G) > c or outputs a tree decomposition of G of width at most 5c in 2O(c)n steps, as a result based on Bodlaender, [1]. But: 8 To get that tw(G) = o(k) for a sole parameter k and graph class G, we need to use our best combinatorics... Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 4/23 ...Enter Light Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 5/23 ...Enter Light Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 5/23 ...Enter Bidimensionality Theorem 1 (Robertson et al. (1994)). Let G be a planar graph with tw(G) ≥ 6l. Then Γl m G. Linear treewidth-grid-minor condition, LTGM Figure: The grid graph Γ10 Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 6/23 ...Enter Bidimensionality Theorem 1 (Robertson et al. (1994)). Let G be a planar graph with tw(G) ≥ 6l. Then Γl m G. Linear treewidth-grid-minor condition, LTGM Figure: The grid graph Γ10 Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 6/23 ...Enter Bidimensionality Theorem 1 (Robertson et al. (1994)). Let G be a planar graph with tw(G) ≥ 6l. Then Γl m G. Linear treewidth-grid-minor condition, LTGM Figure: The grid graph Γ10 Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 6/23 ≥ l2=2 : vc is minor closed! Vertex Cover for grids Figure: vc(Γl) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 ≥ l2=2 : vc is minor closed! Vertex Cover for grids Figure: vc(Γl) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 : vc is minor closed! Vertex Cover for grids 2 Figure: vc(Γl) ≥ l =2 Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 Vertex Cover for grids 2 Figure: vc(Γl) ≥ l =2 : vc is minor closed! Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) planar Vertex Cover and treewidth : vc is minor closed! Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) planar Vertex Cover and treewidth : vc is minor closed! u Let G be planar and vc(G) = k. Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) planar Vertex Cover and treewidth : vc is minor closed! u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 u Therefore, we easily proved tw(G) = Opvc(G) planar Vertex Cover and treewidth : vc is minor closed! u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 planar Vertex Cover and treewidth : vc is minor closed! u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 planar Vertex Cover and treewidth : vc is minor closed! u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l ) So what is Bidimensionality? Take a second look: u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l ) So what is Bidimensionality? Take a second look: u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l ) So what is Bidimensionality? Take a second look: u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l ) So what is Bidimensionality? Take a second look: u Let G be planar and vc(G) = k. u LTGM suggests that Γ tw(G) G 6 u As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 u Therefore, we easily proved tw(G) = Opvc(G) Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al.
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