Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics

Parameterized algorithms are a very useful tool for dealing with NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with problems which are hard to solve even on graphs of bounded tree-width. The drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a new parameter called twin-cover and show that it is capable of solving a wide range of hard problems while also being much less restrictive than vertex cover and attaining low values even on dense graphs. The article begins by introducing a new FPT algorithm for Graph Motif on graphs of bounded vertex cover. This is the first algorithm of this kind for Graph Motif. We continue by defining twin-cover and providing some related results and notions. The next section contains a number of new FPT algorithms on graphs of bounded twin-cover, with a special emphasis on solving problems which are hard even on graphs of bounded tree-width. Finally, section five generalizes the recent results of Michael Lampis for MS1 model checking from vertex cover to twin-cover.

[1]  A. Nijenhuis.  Combinatorial algorithms , 1975 .

[2]  Michael R. Fellows,et al.  What Makes Equitable Connected Partition Easy , 2009, IWPEC.

[3]  Michael Lampis,et al.  Algorithmic Meta-theorems for Restrictions of Treewidth , 2010, Algorithmica.

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

[5]  Jan Kratochvíl.  A Special Planar Satisfiability Problem and a Consequence of Its NP-completeness , 1994, Discret. Appl. Math..

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

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

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

[9]  Cristina G. Fernandes,et al.  Motif Search in Graphs: Application to Metabolic Networks , 2006, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[10]  Mark de Berg,et al.  Algorithms - ESA 2010, 18th Annual European Symposium, Liverpool, UK, September 6-8, 2010. Proceedings, Part I , 2010, ESA.

[11]  Ge Xia,et al.  Improved upper bounds for vertex cover , 2010, Theor. Comput. Sci..

[12]  Michael R. Fellows,et al.  Upper and lower bounds for finding connected motifs in vertex-colored graphs , 2011, J. Comput. Syst. Sci..

[13]  Geevarghese Philip,et al.  On the Kernelization Complexity of Colorful Motifs , 2010, IPEC.

[14]  Udi Rotics,et al.  Clique-Width is NP-Complete , 2009, SIAM J. Discret. Math..

[15]  Saket Saurabh,et al.  Parameterized Algorithms for Boxicity , 2010, ISAAC.

[16]  Robert Ganian.  Thread Graphs, Linear Rank-Width and Their Algorithmic Applications , 2010, IWOCA.

[17]  Michael A. Langston,et al.  Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings , 2006, IWPEC.

[18]  Michael R. Fellows,et al.  On the complexity of some colorful problems parameterized by treewidth , 2011, Inf. Comput..

[19]  D. Berend,et al.  IMPROVED BOUNDS ON BELL NUMBERS AND ON MOMENTS OF SUMS OF RANDOM VARIABLES , 2000 .

[20]  Jirí Fiala,et al.  Parameterized complexity of coloring problems: Treewidth versus vertex cover , 2009, Theor. Comput. Sci..