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
Treedepth Treedepth is a width measure. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 2 / 51
Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51
Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51
Treedepth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 3 / 51
Treedepth 5 3 2 1 6 7 4 8 Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 4 / 51
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
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
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
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
Treedepth Centered Coloring Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 7 / 51
Treedepth Centered Coloring Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 8 / 51
Treedepth Centered Coloring Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 8 / 51
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
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
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
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
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
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
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
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
Treedepth Treedepth t → Maximal path length 2 t − 1. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 13 / 51
Treedepth Treedepth t → Maximal path length 2 t − 1. Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 13 / 51
Treedepth Basic results A DFS is a Treedepth decomposition Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51
Treedepth Basic results A DFS is a Treedepth decomposition Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 14 / 51
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
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
Treedepth Basic results Treedepth to pathwidth Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 15 / 51
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
Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51
Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51
Treedepth Basic results Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 16 / 51
Treedepth Basic results Treedepth by bruteforce G Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51
Treedepth Basic results Treedepth by bruteforce G S Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51
Treedepth Basic results Treedepth by bruteforce S Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51
Treedepth Basic results Treedepth by bruteforce Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51
Treedepth Basic results Treedepth by bruteforce Fernando S´ anchez Villaamil (RWTH) Parameterized Algorithm for Treedepth July 11, 2014 17 / 51
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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