Combinatorial Optimization on Graphs of Bounded Treewidth

There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixed-parameter tractable algorithms. Starting from trees and series-parallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.

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

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

[3]  Hans L. Bodlaender,et al.  On Linear Time Minor Tests with Depth-First Search , 1993, J. Algorithms.

[4]  Georg Gottlob,et al.  Hypertree decompositions and tractable queries , 1998, PODS '99.

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

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

[7]  Rolf Niedermeier,et al.  Improved Fixed-Parameter Algorithms for Two Feedback Set Problems , 2005, WADS.

[8]  Fedor V. Fomin,et al.  Tree decompositions with small cost , 2002, Discret. Appl. Math..

[9]  Bengt Aspvall,et al.  Memory Requirements for Table Computations in Partial \sl k -Tree Algorithms , 2000, Algorithmica.

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

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

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

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

[14]  Harold N. Gabow,et al.  Finding paths and cycles of superpolylogarithmic length , 2004, STOC '04.

[15]  Arie M. C. A. Koster,et al.  Branch and Tree Decomposition Techniques for Discrete Optimization , 2005 .

[16]  Hans L. Bodlaender,et al.  Safe Reduction Rules for Weighted Treewidth , 2002, WG.

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

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

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

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

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

[22]  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..

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

[24]  Georg Gottlob,et al.  Width Parameters Beyond Tree-width and their Applications , 2008, Comput. J..

[25]  Jan Arne Telle,et al.  Algorithms for Vertex Partitioning Problems on Partial k-Trees , 1997, SIAM J. Discret. Math..

[26]  Michael J. Pelsmajer,et al.  Parameterized Algorithms for Feedback Vertex Set , 2004, IWPEC.

[27]  Geoff Whittle,et al.  Matroid tree-width , 2006, Eur. J. Comb..

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

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

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

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

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

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

[34]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

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

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

[37]  Paul D. Seymour,et al.  Graph Minors. XX. Wagner's conjecture , 2004, J. Comb. Theory B.

[38]  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).

[39]  Rolf H. Möhring,et al.  The Pathwidth and Treewidth of Cographs , 1990, SIAM J. Discret. Math..

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

[41]  Hans L. Bodlaender,et al.  Necessary Edges in k-Chordalisations of Graphs , 2003, J. Comb. Optim..

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

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

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

[45]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

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

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

[48]  Fedor V. Fomin,et al.  Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions , 2010, Algorithmica.

[49]  Arie M. C. A. Koster,et al.  Contraction and Treewidth Lower Bounds , 2004, ESA.

[50]  Fedor V. Fomin,et al.  Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Branch Decompositions , 2005, ESA.

[51]  Nobuji Saito,et al.  Linear-time computability of combinatorial problems on series-parallel graphs , 1982, JACM.

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

[53]  Craig A. Tovey,et al.  Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families , 1992, Algorithmica.

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

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

[56]  Erik D. Demaine,et al.  The Bidimensionality Theory and Its Algorithmic Applications , 2008, Comput. J..

[57]  Petr Hliněný,et al.  Matroid tree-width , 2006 .

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

[59]  D. Rose.  A GRAPH-THEORETIC STUDY OF THE NUMERICAL SOLUTION OF SPARSE POSITIVE DEFINITE SYSTEMS OF LINEAR EQUATIONS , 1972 .

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

[61]  Johann Blieberger,et al.  On the Tree Width of Ada Programs , 2004, Ada-Europe.

[62]  Stephen T. Hedetniemi,et al.  Linear Algorithms for Edge-Coloring Trees and Unicyclic Graphs , 1979, Inf. Process. Lett..

[63]  Gregory F. Cooper,et al.  The Computational Complexity of Probabilistic Inference Using Bayesian Belief Networks , 1990, Artif. Intell..

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

[65]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[66]  Bruno Courcelle,et al.  Monadic Second-Order Evaluations on Tree-Decomposable Graphs , 1993, Theor. Comput. Sci..

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

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

[69]  Bengt Aspvall,et al.  Memory Requirements for Table Computations in Partial k-tree Algorithms , 1998, SWAT.

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

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

[72]  Rina Dechter,et al.  Tree Clustering for Constraint Networks , 1989, Artif. Intell..

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

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

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

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

[77]  James R. Lee,et al.  Improved approximation algorithms for minimum-weight vertex separators , 2005, STOC '05.

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

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

[80]  D. Rose.  Triangulated graphs and the elimination process , 1970 .

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

[82]  Yngve Villanger,et al.  Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In , 2006, LATIN.

[83]  Tohru Kikuno,et al.  A linear algorithm for the domination number of a series-parallel graph , 1983, Discret. Appl. Math..

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

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

[86]  Yossi Shiloach,et al.  A Minimum Linear Arrangement Algorithm for Undirected Trees , 1979, SIAM J. Comput..

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

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

[89]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

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

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

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

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

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

[95]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[96]  Michael R. Fellows,et al.  On search decision and the efficiency of polynomial-time algorithms , 1989, STOC '89.