Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time

We give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph G with n vertices, let bw(G) be the branchwidth of G and gm(G) the largest integer g such that G has a gxg grid as a minor. Let c>=1 be a fixed integer and @a,@b arbitrary constants satisfying @a>c+1 and @b>2c+1. We give an algorithm which constructs in O(n^1^+^1^clogn) time a branch-decomposition of G with width at most @abw(G). We also give an algorithm which constructs a gxg grid minor of G with g>=gm(G)@b in O(n^1^+^1^clogn) time. The constants hidden in the Big-O notations are proportional to c@a-(c+1) and c@b-(2c+1), respectively.

[1]  Dimitrios M. Thilikos,et al.  Constructive Linear Time Algorithms for Branchwidth , 1997, ICALP.

[2]  Hisao Tamaki.  A Linear Time Heuristic for the Branch-Decomposition of Planar Graphs , 2003, ESA.

[3]  Alexander Grigoriev,et al.  Treewidth Lower Bounds with Brambles , 2005, Algorithmica.

[4]  Erik D. Demaine,et al.  Bidimensionality, Map Graphs, and Grid Minors , 2005, ArXiv.

[5]  B. A. Reed,et al.  Algorithmic Aspects of Tree Width , 2003 .

[6]  Ken-ichi Kawarabayashi,et al.  Algorithmic graph minor theory: Decomposition, approximation, and coloring , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[7]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[8]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[9]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[10]  Karsten Weihe,et al.  Edge-Disjoint (s, t)-Paths in Undirected Planar Graphs in Linear Time , 1997, J. Algorithms.

[11]  Paul D. Seymour,et al.  Graph minors. X. Obstructions to tree-decomposition , 1991, J. Comb. Theory, Ser. B.

[12]  Jan Kratochvíl,et al.  Computing the branchwidth of interval graphs , 2005, Discret. Appl. Math..

[13]  Robin Thomas,et al.  Call routing and the ratcatcher , 1994, Comb..

[14]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[15]  Milan Ruzic,et al.  Uniform deterministic dictionaries , 2008, TALG.

[16]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[17]  Hisao Tamaki,et al.  Constant-Factor Approximations of Branch-Decomposition and Largest Grid Minor of Planar Graphs in O(n1 + ε) Time , 2009, ISAAC.

[18]  Hisao Tamaki,et al.  Optimal branch-decomposition of planar graphs in O(n3) Time , 2005, TALG.

[19]  Laurent Coupry,et al.  A Simple Linear Algorithm for the Edge-Disjoint (s, t)-Paths Problem in Undirected Planar Graphs , 1997, Inf. Process. Lett..

[20]  Alexander Grigoriev,et al.  Tree-width and large grid minors in planar graphs , 2011, Discret. Math. Theor. Comput. Sci..

[21]  Erik D. Demaine,et al.  Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality , 2005, SODA '05.

[22]  Paul D. Seymour,et al.  Graph minors. I. Excluding a forest , 1983, J. Comb. Theory, Ser. B.

[23]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[24]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[25]  Hisao Tamaki,et al.  Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size , 2010, Algorithmica.

[26]  Dimitrios M. Thilikos,et al.  Catalan structures and dynamic programming in H-minor-free graphs , 2008, SODA '08.

[27]  James R. Lee,et al.  Improved Approximation Algorithms for Minimum Weight Vertex Separators , 2008, SIAM J. Comput..

[28]  Eyal Amir,et al.  Approximation Algorithms for Treewidth , 2010, Algorithmica.

[29]  Rolf Niedermeier.  TREE DECOMPOSITIONS OF GRAPHS , 2006 .