A Parameterized Algorithm for Chordal Sandwich

Given an arbitrary graph G=(V,E) and an additional set of admissible edges F, the Chordal Sandwich problem asks whether there exists a chordal graph (V,E∪F′) such that F′⊆F. This problem arises from perfect phylogeny in evolution and from sparse matrix computations in numerical analysis, and it generalizes the widely studied problems of completions and deletions of arbitrary graphs into chordal graphs. As many related problems, Chordal Sandwich is NP-complete. In this paper we show that the problem becomes tractable when parameterized with a suitable natural measure on the set of admissible edges F. In particular, we give an algorithm with running time $\mathcal{O}(2^{k}n^{5})$ to solve this problem, where k is the size of a minimum vertex cover of the graph (V, F). Hence we show that the problem is fixed parameter tractable when parameterized by k. Note that the parameter does not assume any restriction on the input graph, and it concerns only the additional edge set F.

[1]  Michael R. Fellows,et al.  On problems without polynomial kernels , 2009, J. Comput. Syst. Sci..

[2]  Pinar Heggernes,et al.  Minimal triangulations of graphs: A survey , 2006, Discret. Math..

[3]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

[4]  Martin Charles Golumbic,et al.  Graph Sandwich Problems , 1995, J. Algorithms.

[5]  Fedor V. Fomin,et al.  Exact algorithms for treewidth and minimum fill-in ∗ † , 2006 .

[6]  Fedor V. Fomin,et al.  Finding Induced Subgraphs via Minimal Triangulations , 2009, STACS.

[7]  Hazel Everett,et al.  The Homogeneous Set Sandwich Problem , 1998, Inf. Process. Lett..

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

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

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

[11]  Fedor V. Fomin,et al.  Treewidth computation and extremal combinatorics , 2008, Comb..

[12]  Ioan Todinca,et al.  Treewidth and Minimum Fill-in: Grouping the Minimal Separators , 2001, SIAM J. Comput..

[13]  P. Heggernes,et al.  Computing Minimal Triangulations in Time , 2005 .

[14]  G. Dirac.  On rigid circuit graphs , 1961 .

[15]  Andreas Parra,et al.  Characterizations and Algorithmic Applications of Chordal Graph Embeddings , 1997, Discret. Appl. Math..

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

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

[18]  Peter Buneman,et al.  A characterisation of rigid circuit graphs , 1974, Discret. Math..

[19]  Michael R. Fellows,et al.  Two Strikes Against Perfect Phylogeny , 1992, ICALP.

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

[21]  Michael R. Fellows,et al.  The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs , 2000, Theor. Comput. Sci..

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

[23]  Pinar Heggernes,et al.  Computing Minimal Triangulations in Time O(nalpha log n) = o(n 2.376) , 2005, SIAM J. Discret. Math..

[24]  Dániel Marx.  Chordal Deletion is Fixed-Parameter Tractable , 2008, Algorithmica.

[25]  Martin Charles Golumbic,et al.  Complexity and Algorithms for Graph and Hypergraph Sandwich Problems , 1998, Graphs Comb..

[26]  Celina M. H. de Figueiredo,et al.  On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs , 2007, Theor. Comput. Sci..

[27]  Haim Kaplan,et al.  Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and physical mapping , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[28]  Daniel Lokshtanov.  On the complexity of computing treelength , 2010, Discret. Appl. Math..

[29]  Henning Fernau,et al.  Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves , 2009, STACS.

[30]  Roded Sharan,et al.  A Polynomial Approximation Algorithm for the Minimum Fill-In Problem , 2000, SIAM J. Comput..

[31]  Michel Habib,et al.  Can transitive orientation make sandwich problems easier? , 2007, Discret. Math..

[32]  Michael R. Fellows,et al.  On Problems without Polynomial Kernels (Extended Abstract) , 2008, ICALP.

[33]  Pinar Heggernes,et al.  Faster Parameterized Algorithms for Minimum Fill-In , 2008, ISAAC.

[34]  Maxime Crochemore,et al.  Finding Patterns In Given Intervals , 2007, Fundam. Informaticae.

[35]  M. Steel.  The complexity of reconstructing trees from qualitative characters and subtrees , 1992 .

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

[37]  Celina M. H. de Figueiredo,et al.  The graph sandwich problem for 1-join composition is NP-complete , 2000, Discret. Appl. Math..

[38]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.