Jungles, bundles, and fixed parameter tractability

We give a fixed-parameter tractable (FPT) approximation algorithm computing the path-width of a tournament, and more generally, of a semi-complete digraph. Based on this result, we prove that topological containment and rooted immersion problems are FPT on semi-complete digraphs.

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

[2]  R. Varga,et al.  Proof of Theorem 4 , 1983 .

[3]  Bruno Courcelle,et al.  The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues , 1992, RAIRO Theor. Informatics Appl..

[4]  Jan Obdrzálek,et al.  DAG-width: connectivity measure for directed graphs , 2006, SODA '06.

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

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

[7]  Stephan Kreutzer,et al.  DAG-Width and Parity Games , 2006, STACS.

[8]  Mam Riess Jones.  Color Coding , 1962, Human factors.

[9]  Robin Thomas,et al.  Directed Tree-Width , 2001, J. Comb. Theory, Ser. B.

[10]  Paul D. Seymour,et al.  A well-quasi-order for tournaments , 2011, J. Comb. Theory, Ser. B.

[11]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

[12]  Anders Yeo,et al.  The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments , 2006, Combinatorics, Probability and Computing.

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

[14]  Paul Wollan,et al.  Finding topological subgraphs is fixed-parameter tractable , 2010, STOC '11.

[15]  Martin Grohe,et al.  Computing crossing numbers in quadratic time , 2000, STOC '01.

[16]  Dimitrios M. Thilikos,et al.  Bidimensionality and kernels , 2010, SODA '10.

[17]  Fedor V. Fomin,et al.  Kernels for feedback arc set in tournaments , 2009, J. Comput. Syst. Sci..

[18]  Mamadou Moustapha Kanté,et al.  F-rank-width of (edge-colored) graphs , 2011, CAI 2011.

[19]  Michal Pilipczuk,et al.  Designing FPT Algorithms for Cut Problems Using Randomized Contractions , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[20]  Paul D. Seymour,et al.  Tournament pathwidth and topological containment , 2013, J. Comb. Theory, Ser. B.

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

[22]  Paul D. Seymour,et al.  Tournament immersion and cutwidth , 2012, J. Comb. Theory, Ser. B.

[23]  Saket Saurabh,et al.  Parameterized algorithms for feedback set problems and their duals in tournaments , 2006, Theor. Comput. Sci..

[24]  Robert Ganian,et al.  Clique-width: When Hard Does Not Mean Impossible , 2011, STACS.

[25]  Petr Hliněný,et al.  Are There Any Good Digraph Width Measures? , 2010, IPEC.

[26]  Michael R. Fellows,et al.  Nonconstructive tools for proving polynomial-time decidability , 1988, JACM.

[27]  Noga Alon,et al.  Fast Fast , 2009, ICALP.

[28]  Bruce A. Reed,et al.  Hadwiger's conjecture is decidable , 2009, STOC '09.

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

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

[31]  Mamadou Moustapha Kanté,et al.  The Rank-Width of Edge-Coloured Graphs , 2007, Theory of Computing Systems.

[32]  J. Moon.  On maximal transitive subtournaments , 1971, Proceedings of the Edinburgh Mathematical Society.

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

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

[35]  Patrick Grim,et al.  What is a Contradiction , 2004 .

[36]  Aleksandrs Slivkins.  Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs , 2010, SIAM J. Discret. Math..

[37]  Paul D. Seymour,et al.  Edge-disjoint paths in digraphs with bounded independence number , 2015, J. Comb. Theory, Ser. B.

[38]  Fedor V. Fomin,et al.  Bidimensionality and EPTAS , 2010, SODA '11.

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

[40]  Jørgen Bang-Jensen,et al.  Edge-disjoint in- and out-branchings in tournaments and related path problems , 1991, J. Comb. Theory, Ser. B.

[41]  Carsten Thomassen,et al.  A Polynomial Algorithm for the 2-Path Problem for Semicomplete Digraphs , 1992, SIAM J. Discret. Math..

[42]  Stephan Kreutzer,et al.  Digraph measures: Kelly decompositions, games, and orderings , 2007, SODA '07.

[43]  Felix Schlenk,et al.  Proof of Theorem 3 , 2005 .

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

[45]  Alon Itai,et al.  On the Complexity of Timetable and Multicommodity Flow Problems , 1976, SIAM J. Comput..

[46]  Nir Ailon,et al.  Aggregating inconsistent information: Ranking and clustering , 2008 .

[47]  Aravind Srinivasan,et al.  Splitters and near-optimal derandomization , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

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

[49]  Bruce A. Reed,et al.  A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[50]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs , 2005, JACM.

[51]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[52]  Noga Alon,et al.  Ranking Tournaments , 2006, SIAM J. Discret. Math..

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

[54]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[55]  Maria Chudnovsky,et al.  Disjoint paths in tournaments , 2014, 1411.6226.

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

[57]  Stephan Kreutzer,et al.  Computing excluded minors , 2008, SODA '08.