Logic, graphs, and algorithms

Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all graph properties definable in monadic second-order logi c can be decided in linear time on graphs of bounded tree width. This article is an introduction into the theory underlying s uch meta theorems and a survey of the most important results in this area.

[1]  Martin Grohe.  Generalized Model-Checking Problems for First-Order Logic , 2001, STACS.

[2]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

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

[4]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness II: On Completeness for W[1] , 1995, Theor. Comput. Sci..

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

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

[7]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[8]  Bruno Courcelle,et al.  Linear delay enumeration and monadic second-order logic , 2009, Discret. Appl. Math..

[9]  Sang-il Oum.  Rank-width is less than or equal to branch-width , 2008 .

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

[11]  Sang-il Oum,et al.  Rank‐width is less than or equal to branch‐width , 2008, J. Graph Theory.

[12]  Stephan Kreutzer,et al.  Model Theory Makes Formulas Large , 2007, ICALP.

[13]  Martin Grohe,et al.  An existential locality theorem , 2004, Ann. Pure Appl. Log..

[14]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs , 2004, SODA '04.

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

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

[17]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width , 1998, WG.

[18]  P. Erdös,et al.  Graph Theory and Probability , 1959 .

[19]  Michael R. Fellows,et al.  The Parameterized Complexity of Relational Database Queries and an Improved Characterization of W[1] , 1996, DMTCS.

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

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

[22]  Michael Randolph Garey,et al.  Johnson: "computers and intractability , 1979 .

[23]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[24]  Moshe Y. Vardi.  On the Complexity of Bounded-Variable Queries. , 1995, PODS 1995.

[25]  Paul D. Seymour,et al.  Graph Minors: XVII. Taming a Vortex , 1999, J. Comb. Theory, Ser. B.

[26]  Michael R. Fellows,et al.  FIXED-PARAMETER TRACTABILITY AND COMPLETENESS , 2022 .

[27]  H. Gaifman.  On Local and Non-Local Properties , 1982 .

[28]  Robert E. Tarjan,et al.  Decomposition by clique separators , 1985, Discret. Math..

[29]  Ken-ichi Kawarabayashi,et al.  Approximating the list-chromatic number and the chromatic number in minor-closed and odd-minor-closed classes of graphs , 2006, STOC '06.

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

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

[32]  Martin Grohe,et al.  Definability and Descriptive Complexity on Databases of Bounded Tree-Width , 1999, ICDT.

[33]  Bruno Courcelle.  An Axiomatic Definition of Context-Free Rewriting and its Application to NLC Graph Grammars , 1987, Theor. Comput. Sci..

[34]  Martin Grohe,et al.  Deciding first-order properties of locally tree-decomposable structures , 2000, JACM.

[35]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory B.

[36]  Martin Grohe,et al.  The complexity of first-order and monadic second-order logic revisited , 2004, Ann. Pure Appl. Log..

[37]  Yijia Chen,et al.  On Parameterized Approximability , 2006, IWPEC.

[38]  Ken-ichi Kawarabayashi,et al.  Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner's Contraction , 2006, ISAAC.

[39]  Jörg Flum,et al.  Fixed-Parameter Tractability, Definability, and Model-Checking , 1999, SIAM J. Comput..

[40]  Guillaume Bagan,et al.  MSO Queries on Tree Decomposable Structures Are Computable with Linear Delay , 2006, CSL.

[41]  Stephan Kreutzer,et al.  Locally Excluding a Minor , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[42]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[43]  Leonid Libkin,et al.  Elements of Finite Model Theory , 2004, Texts in Theoretical Computer Science.

[44]  Steven Lindell Haverford.  Computing monadic fixed-points in linear-time on doubly-linked data structures , 2005 .

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

[46]  Jörg Flum,et al.  Query evaluation via tree-decompositions , 2001, JACM.

[47]  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).

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

[49]  R. Steele.  Optimization , 2005 .

[50]  Bruno Courcelle,et al.  Upper bounds to the clique width of graphs , 2000, Discret. Appl. Math..

[51]  Robert E. Tarjan,et al.  Addendum: Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1985, SIAM J. Comput..

[52]  Petr Hlinený,et al.  Finding Branch-Decompositions and Rank-Decompositions , 2007, SIAM J. Comput..

[53]  Markus Frick,et al.  Generalized Model-Checking over Locally Tree-Decomposable Classes , 2002, STACS.

[54]  Stephan Kreutzer,et al.  Approximation Schemes for First-Order Definable Optimisation Problems , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

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

[56]  Paul D. Seymour,et al.  Graph Minors. XVI. Excluding a non-planar graph , 2003, J. Comb. Theory, Ser. B.

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

[58]  Jaroslav Nesetril,et al.  Linear time low tree-width partitions and algorithmic consequences , 2006, STOC '06.

[59]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs X: Linear Orderings , 1996, Theor. Comput. Sci..

[60]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[61]  Jörg Flum,et al.  Mathematical logic , 1985, Undergraduate texts in mathematics.

[62]  Jörg Flum,et al.  Finite model theory , 1995, Perspectives in Mathematical Logic.

[63]  Paul D. Seymour,et al.  Graph minors. III. Planar tree-width , 1984, J. Comb. Theory B.

[64]  Boris Alexander Köpf.  Fixed Parameter Algorithms on Planar Graphs , 2002 .

[65]  Johann A. Makowsky,et al.  Algorithmic uses of the Feferman-Vaught Theorem , 2004, Ann. Pure Appl. Log..

[66]  Bruno Courcelle,et al.  On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic , 2001, Discret. Appl. Math..

[67]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[68]  Paul D. Seymour,et al.  Approximating clique-width and branch-width , 2006, J. Comb. Theory, Ser. B.

[69]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs VII: Graphs as Relational Structures , 1992, Theor. Comput. Sci..

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

[71]  Ittai Abraham,et al.  Compact Routing for Graphs Excluding a Fixed Minor , 2005, DISC.

[72]  Bruno Courcelle,et al.  The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic , 1997, Handbook of Graph Grammars.

[73]  Martin Grohe,et al.  Fixed-parameter tractability and logic , 1999 .

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

[75]  Paul D. Seymour,et al.  Graph minors. XXI. Graphs with unique linkages , 2009, J. Comb. Theory, Ser. B.

[76]  Detlef Seese,et al.  Linear time computable problems and first-order descriptions , 1996, Mathematical Structures in Computer Science.

[77]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion II. Algorithmic aspects , 2008, Eur. J. Comb..

[78]  Arnaud Durand,et al.  First-order queries on structures of bounded degree are computable with constant delay , 2005, TOCL.

[79]  Stephan Kreutzer,et al.  The Expressive Power of Two-Variable Least Fixed-Point Logics , 2005, MFCS.

[80]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion I. Decompositions , 2008, Eur. J. Comb..

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

[82]  Nicole Schweikardt.  On the expressive power of monadic least fixed point logic , 2006, Theor. Comput. Sci..

[83]  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..

[84]  Markus Frick,et al.  Easy instances for model checking , 2001 .

[85]  Paul D. Seymour,et al.  Graph Minors .XIV. Extending an Embedding , 1995, J. Comb. Theory, Ser. B.

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

[87]  James W. Thatcher,et al.  Generalized finite automata theory with an application to a decision problem of second-order logic , 1968, Mathematical systems theory.

[88]  Erik D. Demaine,et al.  Equivalence of local treewidth and linear local treewidth and its algorithmic applications , 2004, SODA '04.

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

[90]  C. Kuratowski.  Sur le problème des courbes gauches en Topologie , 1930 .