Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.

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

[2]  Robert Ganian,et al.  Going Beyond Primal Treewidth for (M)ILP , 2017, AAAI.

[3]  Stefan Kratsch,et al.  Fast Hamiltonicity Checking Via Bases of Perfect Matchings , 2012, J. ACM.

[4]  Christophe Paul,et al.  An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion , 2015, Algorithmica.

[5]  Petr A. Golovach,et al.  Intractability of Clique-Width Parameterizations , 2010, SIAM J. Comput..

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

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

[8]  Robert Ganian,et al.  Solving Problems on Graphs of High Rank-Width , 2015, Algorithmica.

[9]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[10]  Stefan Kratsch,et al.  Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization , 2011, SIAM J. Discret. Math..

[11]  Vít Jelínek.  The rank-width of the square grid , 2010, Discret. Appl. Math..

[12]  Michal Pilipczuk,et al.  Preprocessing subgraph and minor problems: When does a small vertex cover help? , 2012, J. Comput. Syst. Sci..

[13]  Petr A. Golovach,et al.  Clique-width: on the price of generality , 2009, SODA.

[14]  Marcin Pilipczuk,et al.  Finding Hamiltonian Cycle in Graphs of Bounded Treewidth , 2018, SEA.

[15]  Dimitrios M. Thilikos,et al.  (Meta) Kernelization , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[16]  Nobuji Saito,et al.  NP-Completeness of the Hamiltonian Cycle Problem for Bipartite Graphs , 1980 .

[17]  Bruno Courcelle,et al.  Vertex-minors, monadic second-order logic, and a conjecture by Seese , 2007, J. Comb. Theory, Ser. B.

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

[19]  Hans L. Bodlaender,et al.  Vertex Cover Kernelization Revisited , 2010, Theory of Computing Systems.

[20]  Sang-il Oum,et al.  Rank-width and vertex-minors , 2005, J. Comb. Theory, Ser. B.

[21]  Jan Arne Telle,et al.  A unified polynomial-time algorithm for Feedback Vertex Set on graphs of bounded mim-width , 2018, STACS.

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

[23]  Robert Ganian,et al.  Backdoor Treewidth for SAT , 2017, SAT.

[24]  Denis Lapoire,et al.  Recognizability Equals Monadic Second-Order Definability for Sets of Graphs of Bounded Tree-Width , 1998, STACS.

[25]  Dániel Marx,et al.  Slightly superexponential parameterized problems , 2011, SODA '11.

[26]  Robert Ganian,et al.  Meta-kernelization with structural parameters , 2013, J. Comput. Syst. Sci..

[27]  Jan Arne Telle,et al.  Between Treewidth and Clique-Width , 2014, Algorithmica.

[28]  DÁNIEL MARX,et al.  Immersions in Highly Edge Connected Graphs , 2013, SIAM J. Discret. Math..

[29]  M. Vatshelle.  New Width Parameters of Graphs , 2012 .

[30]  Udi Rotics,et al.  Edge dominating set and colorings on graphs with fixed clique-width , 2003, Discret. Appl. Math..

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

[32]  Robert Ganian,et al.  A single-exponential fixed-parameter algorithm for Distance-Hereditary Vertex Deletion , 2016, MFCS.

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

[34]  Robert Ganian,et al.  Combining Treewidth and Backdoors for CSP , 2016, STACS.

[35]  Leizhen Cai,et al.  Parameterized Complexity of Vertex Colouring , 2003, Discret. Appl. Math..

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

[37]  Eun Jung Kim,et al.  A Polynomial Kernel for Distance-Hereditary Vertex Deletion , 2016, Algorithmica.

[38]  Jan Arne Telle,et al.  Maximum Matching Width: New Characterizations and a Fast Algorithm for Dominating Set , 2015, IPEC.

[39]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[40]  Bruno Courcelle,et al.  Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach , 2012, Encyclopedia of mathematics and its applications.

[41]  Robert Ganian,et al.  On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width , 2010, Discret. Appl. Math..

[42]  Michael R. Fellows,et al.  Graph Layout Problems Parameterized by Vertex Cover , 2008, ISAAC.

[43]  Klaus Jansen,et al.  Generalized Coloring for Tree-like Graphs , 1992, WG.

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

[45]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs IV: Definability Properties of Equational Graphs , 1990, Ann. Pure Appl. Log..

[46]  Jakub Gajarský,et al.  Kernelization Using Structural Parameters on Sparse Graph Classes , 2013, ESA.

[47]  Robert Ganian,et al.  A unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width , 2013, Eur. J. Comb..

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

[49]  Egon Wanke,et al.  Deciding Clique-Width for Graphs of Bounded Tree-Width , 2001, J. Graph Algorithms Appl..

[50]  Vít Jelínek.  The Rank-Width of the Square Grid , 2008, WG.

[51]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[52]  Robert Ganian,et al.  Meta-kernelization using well-structured modulators , 2018, Discret. Appl. Math..

[53]  Petr A. Golovach,et al.  Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width , 2014, SIAM J. Comput..

[54]  Stefan Kratsch,et al.  Fast Hamiltonicity Checking Via Bases of Perfect Matchings , 2018, J. ACM.

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