Not So Easy Problems for Tree Decomposable Graphs

We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some recent results for problems regarding list and equitable colorings, general factors, and generalized satisfiability. Second we establish a new hardness result for the problem of minimizing the maximum weighted outdegree for orientations of edge-weighted graphs of bounded treewidth.

[1]  S. Janson,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

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

[3]  Stefan Szeider,et al.  On Fixed-Parameter Tractable Parameterizations of SAT , 2003, SAT.

[4]  Jirí Fiala,et al.  Distance Constrained Labelings of Graphs of Bounded Treewidth , 2005, ICALP.

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

[6]  J. Schiff.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[7]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

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

[9]  Alexandr V. Kostochka,et al.  On Equitable Coloring of d-Degenerate Graphs , 2005, SIAM J. Discret. Math..

[10]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[11]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[12]  Stefan Szeider,et al.  Monadic second order logic on graphs with local cardinality constraints , 2008, TOCL.

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

[14]  Marko Samer,et al.  Constraint satisfaction with bounded treewidth revisited , 2006, J. Comput. Syst. Sci..

[15]  Michael R. Fellows,et al.  On the Complexity of Some Colorful Problems Parameterized by Treewidth , 2007, COCOA.

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

[17]  Béla Bollobás,et al.  List-colourings of graphs , 1985, Graphs Comb..

[18]  Ko-Wei Lih,et al.  Equitable Coloring of Graphs , 1998 .

[19]  Krzysztof Pietrzak,et al.  On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems , 2003, J. Comput. Syst. Sci..

[20]  Fedor V. Fomin,et al.  Equitable Colorings of Bounded Treewidth Graphs , 2004, MFCS.

[21]  Zsolt Tuza,et al.  Graph colorings with local constraints - a survey , 1997, Discuss. Math. Graph Theory.

[22]  David S. Johnson,et al.  COMPLEXITY RESULTS FOR BANDWIDTH MINIMIZATION , 1978 .

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

[24]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[25]  Ton Kloks.  Treewidth, Computations and Approximations , 1994, Lecture Notes in Computer Science.

[26]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[27]  L. Lovász.  The factorization of graphs. II , 1972 .

[28]  Marko Samer,et al.  Tractable cases of the extended global cardinality constraint , 2009, Constraints.

[29]  Georg Gottlob,et al.  Fixed-parameter complexity in AI and nonmonotonic reasoning , 1999, Artif. Intell..

[30]  Gérard Cornuéjols,et al.  General factors of graphs , 1988, J. Comb. Theory, Ser. B.

[31]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[32]  R. Baker.  Factorization of graphs , 1975 .

[33]  Marko Samer,et al.  Algorithms for propositional model counting , 2007, J. Discrete Algorithms.

[34]  Yuichi Asahiro,et al.  Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree , 2008, Discret. Appl. Math..