Maximum Common Subgraph: Upper Bound and Lower Bound Results

In bio informatics, the biological structure matching problems can be formulated as the problem of finding the maximum common subgraph. Among the many different variants of the maximum common subgraph problem, the maximum common induced subgraph of two graphs is of special interest. In this paper, first we derive the lower bound result for the exact algorithms of the maximum common induced subgraph of two graphs, based on the current research progress in the area of parameterized computation. Then we investigate the upper bound and design the approaches for addressing this problem. Basically, this problem can be reduced to find a maximum clique in the product graph of the two given graphs. In consideration of the upper bound result, the obtained lower bound result is asymptotically tight. We would like to point out that our lower bound result presented here is currently best-known

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

[2]  Ge Xia,et al.  Tight lower bounds for certain parameterized NP-hard problems , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[3]  Mihalis Yannakakis,et al.  On the Complexity of Database Queries , 1999, J. Comput. Syst. Sci..

[4]  Michael R. Fellows,et al.  Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems , 2003, CATS.

[5]  Russell L. Malmberg,et al.  Efficient Parameterized Algorithm for Biopolymer Structure-Sequence Alignment , 2005, WABI.

[6]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[7]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[8]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 2001, J. Algorithms.

[9]  Bin Ma,et al.  A PTAS for Distinguishing (Sub)string Selection , 2002, ICALP.

[10]  Xiuzhen Huang,et al.  Parameterized complexity and polynomial-time approximation schemes , 2004 .

[11]  Peter Willett,et al.  Maximum common subgraph isomorphism algorithms for the matching of chemical structures , 2002, J. Comput. Aided Mol. Des..

[12]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[13]  Nagiza F. Samatova,et al.  A New Approach and Faster Exact Methods for the Maximum Common Subgraph Problem , 2005, COCOON.

[14]  Svetha Venkatesh,et al.  Video indexing and similarity retrieval by largest common subgraph detection using decision trees , 2001, Pattern Recognit..

[15]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[16]  Temple F. Smith,et al.  Analysis and algorithms for protein sequence–structure alignment , 1998 .

[17]  D. Eisenberg,et al.  A method to identify protein sequences that fold into a known three-dimensional structure. , 1991, Science.

[18]  S. Bryant,et al.  Statistics of sequence-structure threading. , 1995, Current opinion in structural biology.

[19]  Liming Cai,et al.  On Fixed-Parameter Tractability and Approximability of NP Optimization Problems , 1997, J. Comput. Syst. Sci..

[20]  Elena Rivas,et al.  Noncoding RNA gene detection using comparative sequence analysis , 2001, BMC Bioinformatics.

[21]  Ge Xia,et al.  W-Hardness Under Linear FPT-Reductions: Structural Properties and Further Applications , 2005, COCOON.

[22]  S. Eddy,et al.  tRNAscan-SE: a program for improved detection of transfer RNA genes in genomic sequence. , 1997, Nucleic acids research.

[23]  Bin Ma,et al.  Distinguishing string selection problems , 2003, SODA '99.

[24]  Mihalis Yannakakis,et al.  On limited nondeterminism and the complexity of the V-C dimension , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[25]  Ronald L. Rivest,et al.  Introduction to Algorithms, Second Edition , 2001 .

[26]  Michael Trevor Hallett.  An integrated complexity analysis of problems from computational biology , 1998 .

[27]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[28]  Ying Xu,et al.  Raptor: Optimal Protein Threading by Linear Programming , 2003, J. Bioinform. Comput. Biol..

[29]  Tao Jiang,et al.  On the Approximation of Shortest Common Supersequences and Longest Common Subsequences , 1995, SIAM J. Comput..

[30]  Iyad A. Kanj,et al.  Tight lower bounds for certain parameterized NP-hard problems , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[31]  Bin Ma,et al.  Genetic Design of Drugs Without Side-Effects , 2003, SIAM J. Comput..

[32]  Ge Xia,et al.  Linear FPT reductions and computational lower bounds , 2004, STOC '04.

[33]  Ulrike Stege,et al.  Solving large FPT problems on coarse-grained parallel machines , 2003, J. Comput. Syst. Sci..

[34]  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..

[35]  S. Eddy.  Computational Genomics of Noncoding RNA Genes , 2002, Cell.

[36]  Jianer Chen,et al.  Integrating Sample-Driven and Pattern-Driven Approaches in Motif Finding , 2004, WABI.

[37]  Russell L. Malmberg,et al.  Efficient Parameterized Algorithm for Biopolymer Structure-Sequence Alignment , 2005, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[38]  Viggo Kann,et al.  On the Approximability of the Maximum Common Subgraph Problem , 1992, STACS.

[39]  Radu Horaud,et al.  Stereo Correspondence Through Feature Grouping and Maximal Cliques , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[40]  Bin Ma,et al.  On the closest string and substring problems , 2002, JACM.

[41]  Ying Xu,et al.  An Efficient Computational Method for Globally Optimal Threading , 1998, J. Comput. Biol..

[42]  Rolf Niedermeier,et al.  2 00 2 Parameterized Intractability of Motif Search Problems ∗ , 2002 .

[43]  Michael R. Fellows,et al.  The Parameterized Complexity of Sequence Alignment and Consensus , 1994, CPM.

[44]  Michael R. Fellows,et al.  Parameterized complexity analysis in computational biology , 1995, Comput. Appl. Biosci..

[45]  J. Doudna.  Structural genomics of RNA , 2000, Nature Structural Biology.