Parameterized and Exact Computation

We study parameterized enumeration problems where we are interested in all solutions of limited size, rather than just some minimum solution. In particular, we study the computation of the transversal hypergraph restricted to hyperedges with at most k elements. Then we apply the results and techniques to almost-perfect phylogeny reconstruction in computational biology. We also derive certain concise descriptions of all vertex covers of size at most k in a graph, within less than the trivial time bound.

[1]  B. Bollobás.  On generalized graphs , 1965 .

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

[3]  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..

[4]  Dániel Marx,et al.  Precoloring extension on unit interval graphs , 2006, Discret. Appl. Math..

[5]  Dániel Marx,et al.  Precoloring Extension on Chordal Graphs , 2006 .

[6]  Krzysztof Pietrzak,et al.  On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems , 2003, J. Comput. Syst. Sci..

[7]  Detlef Seese,et al.  The Structure of Models of Decidable Monadic Theories of Graphs , 1991, Ann. Pure Appl. Log..

[8]  Noga Alon,et al.  Finding and counting given length cycles , 1997, Algorithmica.

[9]  Zsolt Tuza,et al.  Precoloring extension. I. Interval graphs , 1992, Discret. Math..

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

[11]  Leizhen Cai,et al.  Fixed-Parameter Tractability of Graph Modification Problems for Hereditary Properties , 1996, Inf. Process. Lett..

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

[13]  Z. Tuza,et al.  PRECOLORING EXTENSION. II. GRAPHS CLASSES RELATED TO BIPARTITE GRAPHS , 1993 .

[14]  Stefan Szeider,et al.  Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable , 2003, COCOON.

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

[16]  P. Seymour,et al.  Surveys in combinatorics 1985: Graph minors – a survey , 1985 .

[17]  Zsolt Tuza,et al.  Precoloring Extension III: Classes of Perfect Graphs , 1996, Combinatorics, Probability and Computing.

[18]  Haim Kaplan,et al.  Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs , 1999, SIAM J. Comput..

[19]  John Michael Robson,et al.  Algorithms for Maximum Independent Sets , 1986, J. Algorithms.

[20]  B. Monien.  How to Find Long Paths Efficiently , 1985 .

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