1. On the complexity of finding common approximate substrings Theoretical Computer Science 306 (2003) 407-430 Patricia A. Evans, Andrew D. Smith, H. Todd Wareham Presentation by Valerie Hajdik 4/28/2006

2. Presentation overview Common Approximate Substring (CAS) Fixed-parameter tractability Parameterized complexity classes CAS(l, Σ ) is in FPT Variants of CAS

3. Motivation for studying CAS Related to problems in computational biology Motif-finding, multiple sequence alignment Lower bounds for CAS could imply lower bounds for other problems

5. Example of a center string of length 5 with d = 1

6. Parameterized Computation Sometimes NP-hard problems can be solved quickly in practice Some “aspect” of the problem (called the parameter) is bounded fixed-parameter tractable:“ “? can be decided in running time

7. Parameterized Complexity Classes A problem is in FPT if it is fixed-parameter tractable A problem is not in FPT if it is hard for any of {W[1], W[2],…W[P],…XP } A problem belongs to W[P] if it parametrically reduces to WCS t,h * * WCS t,h is WEIGHTED WEFT t DEPTH h CIRCUIT SATISFIABILITY

8. Many problems have natural parameters that are small in practice Allows us to get exact, efficient solutions to some NP-hard problems Toolkit of algorithmic techniques for FPT problems makes algorithm design easier Why should you care?

9. CAS(l, Σ ) is in FPT Algorithm 1: Generate all possible strings of length l over Σ and check each one to see if it’s a center | Σ | l strings O(mnl) time to check each Algorithm 1 runs in O(| Σ | l mnl) time m = # of strings, n = length of each string, l = length of center string

10. Parameterized Complexity of CAS with different parameters

11. Variants in FPT m = # of strings, n = length of each string, l = length of center string d = Hamming distance

12. Homework Problem