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A Faster Parameterized Algorithm for Treedepth Felix Reidl, Peter Rossmanith, Fernando S anchez Villaamil Somnath Sikdar RWTH Aachen University July 11, 2014 Fernando S anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July

  1. A Faster Parameterized Algorithm for Treedepth Felix Reidl, Peter Rossmanith, Fernando S´ anchez Villaamil Somnath Sikdar RWTH Aachen University July 11, 2014 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 1 / 51

  2. Treedepth Treedepth is a width measure. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 2 / 51

  3. Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51

  4. Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51

  5. Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51

  6. Treedepth 5 3 2 1 6 7 4 8 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51

  7. Treedepth 5 3 1 2 1 2 3 4 6 7 6 5 7 8 4 8 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51

  8. Treedepth 5 3 1 2 1 2 3 4 6 7 6 5 7 8 4 8 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51

  9. Treedepth Definition (Treedepth decomposition) A treedepth decomposition of a graph G is a rooted forest F such that V ( G ) ⊆ V ( F ) and E ( G ) ⊆ E ( clos ( F )). Definition (Treedepth) The treedepth td ( G ) of a graph G is the minimum height of any treedepth decomposition of G . Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 5 / 51

  10. Treedepth A strange width measure... “So many choices” —Dr. Dre A graph G has treedepth at most t if G is a subgraph the closure of a tree (forest) of height ≤ t G has a centered coloring with t colors G has a ranked coloring with t colors G is the subgraph of a trivially perfect graph with clique size ≤ t Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 6 / 51

  11. Treedepth Centered Coloring Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 7 / 51

  12. Treedepth Centered Coloring Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 8 / 51

  13. Treedepth Centered Coloring Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 8 / 51

  14. Treedepth Ranked Coloring 5 1 1 4 2 2 3 1 1 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 9 / 51

  15. Treedepth Ranked Coloring 5 1 1 5 4 4 2 2 2 3 2 3 1 1 1 1 1 1 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 9 / 51

  16. Treedepth Trivially Perfect Graphs G is the subgraph of a trivially perfect graph with clique size at most t . Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 10 / 51

  17. Treedepth Trivially Perfect Graphs G is the subgraph of a trivially perfect graph with clique size at most t . Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 10 / 51

  18. Treedepth Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇ setˇ ril and Ossona de Mendez [2008] Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 11 / 51

  19. Treedepth Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇ setˇ ril and Ossona de Mendez [2008] Related to... layouting of VLSI chips star height of regular languages characterizing bounded expansion graph classes counting subgraphs [New results coming] Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 11 / 51

  20. Treedepth Arises again and again Introduced as... minimum elimination tree by Pothen [1988] ordered coloring by Katchalski et al. [1995] vertex ranking by Bodlaender et al. [1998] again as treedepth by Neˇ setˇ ril and Ossona de Mendez [2008] Related to... layouting of VLSI chips star height of regular languages characterizing bounded expansion graph classes counting subgraphs [New results coming] Personal opinion : Treedepth is the most useful definition. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 11 / 51

  21. Treedepth 4 1 2 3 4 3 5 6 5 7 1 2 7 6 Treedepth t → Maximal path length 2 t − 1. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 12 / 51

  22. Treedepth Treedepth t → Maximal path length 2 t − 1. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 13 / 51

  23. Treedepth Treedepth t → Maximal path length 2 t − 1. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 13 / 51

  24. Treedepth Basic results A DFS is a Treedepth decomposition Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51

  25. Treedepth Basic results A DFS is a Treedepth decomposition Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51

  26. Treedepth Basic results A DFS is a Treedepth decomposition Treedepth t ⇒ Maximal path length 2 t − 1 ⇒ 2 t -approximation Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51

  27. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  28. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  29. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  30. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  31. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  32. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  33. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  34. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  35. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  36. Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  37. Treedepth Basic results Treedepth to pathwidth tw ( G ) ≤ pw ( G ) ≤ td ( G ) − 1 Treedepth t ⇒ Path decomposition of width 2 t − 2 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51

  38. Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51

  39. Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51

  40. Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51

  41. Treedepth Basic results Treedepth by bruteforce G Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

  42. Treedepth Basic results Treedepth by bruteforce G S Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

  43. Treedepth Basic results Treedepth by bruteforce S Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

  44. Treedepth Basic results Treedepth by bruteforce Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

  45. Treedepth Basic results Treedepth by bruteforce Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51

  46. Treedepth Basic results Parameterized algorithms Open problem by Neˇ setˇ ril and Ossona de Mendez [2012] Is there a simple linear time algorithm to check td ( G ) ≤ t for fixed t ? Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 18 / 51

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