Fast approximation schemes for K3, 3-minor-free or K5-minor-free graphs

As the class of graphs of bounded treewidth is of limited size, we need to solve NP-hard problems for wider classes of graphs than this class. Eppstein introduced a new concept which can be considered as a generalization of bounded treewidth. A graph G has {em locally bounded treewidth} if for each vertex v of G, the treewidth of the subgraph of G induced on all vertices of distance at most r from v is only a function of r, called local treewidth. So far the only graphs determined to have small local treewidth are planar graphs. In this paper, we prove that any graph excluding one of K_{5} or K_{3,3} as a minor has local treewidth bounded by 3k+4. As a result, we can design practical polynomial-time approximation schemes for both minimization and maximization problems on these classes of non-planar graphs.

[1]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[2]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[3]  Rolf Niedermeier,et al.  Fixed Parameter Algorithms for PLANAR DOMINATING SET and Related Problems , 2000, SWAT.

[4]  Martin Grohe.  Local Tree-Width, Excluded Minors, and Approximation Algorithms , 2003, Comb..

[5]  Jens Lagergren,et al.  Efficient Parallel Algorithms for Graphs of Bounded Tree-Width , 1996, J. Algorithms.

[6]  Norishige Chiba,et al.  An Approximation Algorithm for the Maximum Independent Set Problem on Planar Graphs , 1982, SIAM J. Comput..

[7]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[8]  Donald J. ROSE,et al.  On simple characterizations of k-trees , 1974, Discret. Math..

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

[10]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[11]  Mihalis Yannakakis,et al.  Node-and edge-deletion NP-complete problems , 1978, STOC.

[12]  Bruno Courcelle,et al.  An algebraic theory of graph reduction , 1993, JACM.

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  David Eppstein.  Diameter and Treewidth in Minor-Closed Graph Families , 2000, Algorithmica.

[15]  Rolf H. Möhring,et al.  The Pathwidth and Treewidth of Cographs , 1993, SIAM J. Discret. Math..

[16]  Dimitrios M. Thilikos,et al.  Treewidth for Graphs with Small Chordality , 1997, Discret. Appl. Math..

[17]  S. Arnborg,et al.  Characterization and recognition of partial 3-trees , 1986 .

[18]  Zhi-Zhong Chen.  Efficient Approximation Schemes for Maximization Problems on K3,3-free or K5-free Graphs , 1998, J. Algorithms.

[19]  C. Pandu Rangan,et al.  Treewidth of Circular-Arc Graphs , 1994, SIAM J. Discret. Math..

[20]  K. Wagner.  Über eine Eigenschaft der ebenen Komplexe , 1937 .

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

[22]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[23]  Hans L. Bodlaender,et al.  Treewidth: Algorithmic Techniques and Results , 1997, MFCS.

[24]  Jan Arne Telle,et al.  Practical Algorithms on Partial k-Trees with an Application to Domination-like Problems , 1993, WADS.

[25]  Daniel P. Sanders.  On Linear Recognition of Tree-Width at Most Four , 1996, SIAM J. Discret. Math..

[26]  Hans L. Bodlaender,et al.  Reduction Algorithms for Constructing Solutions in Graphs with Small Treewidth , 1996, COCOON.

[27]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[28]  Paul D. Seymour,et al.  Excluding a graph with one crossing , 1991, Graph Structure Theory.

[29]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[30]  Detlef Seese,et al.  Problems Easy for Tree-Decomposable Graphs (Extended Abstract) , 1988, ICALP.

[31]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[32]  Ton Kloks,et al.  Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs , 1993, J. Algorithms.

[33]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[34]  Xin He,et al.  NC Algorithms for Partitioning Planar Graphs into Induced Forests and Approximating NP-Hard Problems , 1995, WG.

[35]  Robin Thomas,et al.  A separator theorem for graphs with an excluded minor and its applications , 1990, STOC '90.

[36]  David Eppstein,et al.  The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems , 1994, GD.

[37]  John R. Gilbert,et al.  Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree , 1995, J. Algorithms.

[38]  Dieter Kratsch,et al.  Treewidth and Pathwidth of Permutation Graphs , 1993, ICALP.

[39]  Ton Kloks,et al.  Treewidth of Circle Graphs , 1993, ISAAC.

[40]  Takao Asano,et al.  An Approach to the Subgraph Homeomorphism Problem , 1985, Theor. Comput. Sci..

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

[42]  Andrew M. Peck,et al.  Partitioning Planar Graphs , 1992, SIAM J. Comput..

[43]  Zhi-Zhong Chen.  NC Algorithms for Partitioning Sparse Graphs into Induced Forests with an Application , 1995, ISAAC.

[44]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.