Discovering Treewidth

Treewidth is a graph parameter with several interesting theoretical and practical applications. This survey reviews algorithmic results on determining the treewidth of a given graph, and finding a tree decomposition of small width. Both theoretical results, establishing the asymptotic computational complexity of the problem, as experimental work on heuristics (both for upper bounds as for lower bounds), preprocessing, exact algorithms, and postprocessing are discussed.

[1]  Dan Geiger,et al.  A Practical Algorithm for Finding Optimal Triangulations , 1997, AAAI/IAAI.

[2]  Rajeev Raman,et al.  Algorithms — ESA 2002 , 2002, Lecture Notes in Computer Science.

[3]  Hans L. Bodlaender,et al.  New Upper Bound Heuristics for Treewidth , 2005, WEA.

[4]  Arie M. C. A. Koster,et al.  Safe Reduction Rules for Weighted Treewidth , 2002, Algorithmica.

[5]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

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

[7]  M. Habib,et al.  Treewidth of cocomparability graphs and a new order-theoretic parameter , 1994 .

[8]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[9]  Paul D. Seymour,et al.  Tour Merging via Branch-Decomposition , 2003, INFORMS J. Comput..

[10]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width , 2000, Theory of Computing Systems.

[11]  Mikkel Thorup,et al.  All Structured Programs have Small Tree-Width and Good Register Allocation , 1998, Inf. Comput..

[12]  Ioan Todinca,et al.  Treewidth and Minimum Fill-in: Grouping the Minimal Separators , 2001, SIAM J. Comput..

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

[14]  Gerhard J. Woeginger,et al.  Exact Algorithms for NP-Hard Problems: A Survey , 2001, Combinatorial Optimization.

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

[16]  Illya V. Hicks.  Graphs, branchwidth, and tangles! Oh my! , 2005, Networks.

[17]  Arie M. C. A. Koster,et al.  Solving partial constraint satisfaction problems with tree decomposition , 2002, Networks.

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

[19]  Pedro Larrañaga,et al.  Decomposing Bayesian networks: triangulation of the moral graph with genetic algorithms , 1997, Stat. Comput..

[20]  Kristian G. Olesen,et al.  Maximal Prime Subgraph Decomposition of Bayesian Networks , 2001, FLAIRS.

[21]  Fedor V. Fomin,et al.  Exact (Exponential) Algorithms for Treewidth and Minimum Fill-In , 2004, ICALP.

[22]  Mikkel Thorup,et al.  Structured Programs have Small Tree-Width and Good Register Allocation (Extended Abstract) , 1997, WG.

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

[24]  Rina Dechter,et al.  Bucket Elimination: A Unifying Framework for Reasoning , 1999, Artif. Intell..

[25]  Pinar Heggernes,et al.  Efficient Implementation of a Minimal Triangulation Algorithm , 2002, ESA.

[26]  Dieter Kratsch,et al.  A Generalization of AT-Free Graphs and a Generic Algorithm for Solving Triangulation Problems , 2001, Algorithmica.

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

[28]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[29]  Thomas Wolle,et al.  Contraction Degeneracy on Cographs , 2004 .

[30]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

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

[32]  S. Arnborg,et al.  Finding Minimal Forbidden Minors Using a Finite Congruence , 1991, ICALP.

[33]  Dan Geiger,et al.  A sufficiently fast algorithm for finding close to optimal clique trees , 2001, Artif. Intell..

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

[35]  Siddharthan Ramachandramurthi,et al.  The Structure and Number of Obstructions to Treewidth , 1997, SIAM J. Discret. Math..

[36]  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).

[37]  Elias Dahlhaus,et al.  Minimal Elimination Ordering Inside a Given Chordal Graph , 1997, WG.

[38]  Bruce A. Reed,et al.  Finding approximate separators and computing tree width quickly , 1992, STOC '92.

[39]  Illya V. Hicks.  Branch decompositions and their applications , 2000 .

[40]  Robin Thomas,et al.  Graph Searching and a Min-Max Theorem for Tree-Width , 1993, J. Comb. Theory, Ser. B.

[41]  Elias Dahlhaus.  Minimum Fill-in and Treewidth for Graphs Modularly Decomposable into Chordal Graphs , 1998, WG.

[42]  Susanne Albers,et al.  Algorithms – ESA 2004 , 2004, Lecture Notes in Computer Science.

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

[44]  David R. Wood,et al.  Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing , 2002, FSTTCS.

[45]  Vibhav Gogate,et al.  A Complete Anytime Algorithm for Treewidth , 2004, UAI.

[46]  Eyal Amir,et al.  Efficient Approximation for Triangulation of Minimum Treewidth , 2001, UAI.

[47]  Udi Rotics,et al.  Computing the Treewidth and the Minimum Fill-in with the Modular Decomposition , 2002, SWAT.

[48]  Barry W. Peyton,et al.  Maximum Cardinality Search for Computing Minimal Triangulations of Graphs , 2004, Algorithmica.

[49]  Jacques Carlier,et al.  New Lower and Upper Bounds for Graph Treewidth , 2003, WEA.

[50]  Bruno Courcelle,et al.  Graph grammars, monadic second-order logic and the theory of graph minors , 1991, Graph Structure Theory.

[51]  Arie M. C. A. Koster,et al.  Frequency assignment : models and algorithms , 1999 .

[52]  Carsten Thomassen,et al.  Highly Connected Sets and the Excluded Grid Theorem , 1999, J. Comb. Theory, Ser. B.

[53]  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.

[54]  Arie M. C. A. Koster,et al.  Contraction and Treewidth Lower Bounds , 2004, J. Graph Algorithms Appl..

[55]  David J. Spiegelhalter,et al.  Local computations with probabilities on graphical structures and their application to expert systems , 1990 .

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

[57]  Andreas Parra,et al.  Characterizations and Algorithmic Applications of Chordal Graph Embeddings , 1997, Discret. Appl. Math..

[58]  Dieter Kratsch,et al.  Treewidth and Minimum Fill-in on d-Trapezoid Graphs , 1998, J. Graph Algorithms Appl..

[59]  Dieter Kratsch,et al.  On treewidth approximations , 2004, Discret. Appl. Math..

[60]  Siddharthan Ramachandramurthi,et al.  A Lower Bound for Treewidth and Its Consequences , 1994, WG.

[61]  Hans L. Bodlaender.  Improved Self-reduction Algorithms for Graphs with Bounded Treewidth , 1994, Discret. Appl. Math..

[62]  Bruce A. Reed,et al.  An Improved Algorithm for Finding Tree Decompositions of Small Width , 1999, WG.

[63]  Ioan Todinca,et al.  Listing all potential maximal cliques of a graph , 2000, Theor. Comput. Sci..

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

[65]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

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

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

[68]  D. R. Lick,et al.  k-Degenerate Graphs , 1970, Canadian Journal of Mathematics.

[69]  R. Diestel.  Graph Decompositions: A Study in Infinite Graph Theory , 1990 .

[70]  Erik D. Demaine,et al.  Bidimensionality: new connections between FPT algorithms and PTASs , 2005, SODA '05.

[71]  Linda C. van der Gaag,et al.  Pre-processing for Triangulation of Probabilistic Networks , 2001, UAI.

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

[73]  Rolf Niedermeier,et al.  Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs , 2005, Discret. Appl. Math..

[74]  Illya V. Hicks,et al.  Planar Branch Decompositions I: The Ratcatcher , 2005, INFORMS J. Comput..

[75]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[76]  Arie M. C. A. Koster,et al.  Treewidth: Computational Experiments , 2001, Electron. Notes Discret. Math..

[77]  Arie M. C. A. Koster,et al.  Safe separators for treewidth , 2006, Discret. Math..

[78]  Éva Tardos,et al.  Algorithm design , 2005 .

[79]  Erik D. Demaine,et al.  Approximation algorithms for classes of graphs excluding single-crossing graphs as minors , 2004, J. Comput. Syst. Sci..

[80]  Ioan Todinca,et al.  Approximating the treewidth of AT-free graphs , 2003, Discret. Appl. Math..

[81]  Klaus Jansen,et al.  Experimental and Efficient Algorithms , 2003, Lecture Notes in Computer Science.

[82]  Eugene L. Lawler,et al.  The recognition of Series Parallel digraphs , 1979, SIAM J. Comput..

[83]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

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

[85]  G. Grätzer,et al.  The free $$ \mathfrak{m} $$ -lattice on the poset H , 1984 .

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

[87]  Jacques Carlier,et al.  Heuristic and metaheuristic methods for computing graph treewidth , 2004, RAIRO Oper. Res..

[88]  Arie M. C. A. Koster,et al.  On the maximum cardinality search lower bound for treewidth , 2004, Discret. Appl. Math..

[89]  Atsuko Yamaguchi,et al.  Graph Complexity of Chemical Compounds in Biological Pathways , 2003 .

[90]  P. Heggernes,et al.  Computing minimal triangulations in time O(nα log n) = o(n2.376) , 2005, SODA '05.

[91]  G. Szekeres,et al.  An inequality for the chromatic number of a graph , 1968 .

[92]  Dieter Kratsch,et al.  Treewidth of Chordal Bipartite Graphs , 1993, J. Algorithms.

[93]  Daniel Bienstock,et al.  Graph Searching, Path-Width, Tree-Width and Related Problems (A Survey) , 1989, Reliability Of Computer And Communication Networks.

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

[95]  Uffe Kjærulff.  Optimal decomposition of probabilistic networks by simulated annealing , 1992 .

[96]  Illya V. Hicks.  Planar Branch Decompositions II: The Cycle Method , 2005, INFORMS J. Comput..

[97]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

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

[99]  Arie M. C. A. Koster,et al.  Safe Seperators for Treewidth , 2004, ALENEX/ANALC.

[100]  R. Möhring.  Algorithmic Aspects of Comparability Graphs and Interval Graphs , 1985 .

[101]  Arie M. C. A. Koster,et al.  Degree-Based Treewidth Lower Bounds , 2005, WEA.

[102]  Rolf Niedermeier,et al.  Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs , 2002, Algorithmica.

[103]  Pinar Heggernes,et al.  Computing Minimal Triangulations in Time O(nalpha log n) = o(n 2.376) , 2005, SIAM J. Discret. Math..

[104]  Illya V. Hicks,et al.  Branch decompositions and minor containment , 2004, Networks.

[105]  Giuseppe Liotta.  Graph Algorithms And Applications 2 , 2004 .

[106]  R. Fildes.  Journal of the Royal Statistical Society (B): Gary K. Grunwald, Adrian E. Raftery and Peter Guttorp, 1993, “Time series of continuous proportions”, 55, 103–116.☆ , 1993 .

[107]  Torben Hagerup,et al.  Parallel Algorithms with Optimal Speedup for Bounded Treewidth , 1995, SIAM J. Comput..

[108]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[109]  Brian Lucena,et al.  A New Lower Bound for Tree-Width Using Maximum Cardinality Search , 2003, SIAM J. Discret. Math..

[110]  Frank Harary,et al.  Graph Theory , 2016 .

[111]  Jens Gustedt,et al.  The Treewidth of Java Programs , 2002, ALENEX.

[112]  Pinar Heggernes,et al.  A practical algorithm for making filled graphs minimal , 2001, Theor. Comput. Sci..

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

[114]  David M. Mount,et al.  Algorithm Engineering and Experiments , 2002, Lecture Notes in Computer Science.

[115]  Albert Mercade.  Branch decompositions and minor containment , 2004 .

[116]  Michael Jünger,et al.  Combinatorial optimization - Eureka, you shrink! , 2003 .

[117]  Hans L. Bodlaender,et al.  Parallel Algorithms for Series Parallel Graphs and Graphs with Treewidth Two1 , 2001, Algorithmica.

[118]  Jeremy P. Spinrad,et al.  On Treewidth and Minimum Fill-In of Asteroidal Triple-Free Graphs , 1997, Theor. Comput. Sci..

[119]  Ton Kloks,et al.  A Linear Time Algorithm for Minimum Fill-in and Treewidth for Distance Hereditary Graphs , 2000, Discret. Appl. Math..

[120]  Robin Thomas,et al.  Algorithms Finding Tree-Decompositions of Graphs , 1991, J. Algorithms.