Brona Brejova, Daniel G. Brown, Ian M. Harrower, Alejandro Lopez-Ortiz, Tomas Vinar. Sharper Upper and Lower Bounds for an Approximation Scheme for CONSENSUS-PATTERN. In Alberto Apostolico, Maxime Crochemore, Kunsoo Park, ed., Combinatorial Pattern Matching, 16th Annual Symposium (CPM), 3537 volume of Lecture Notes in Computer Science, pp. 1-10, Jeju Island, Korea, June 19-22 2005. Springer.
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We present sharper upper and lower bounds for a known polynomial-time approximation scheme due to Li, Ma and Wang 2002 for the CONSENSUS-PATTERN problem. This NP-hard problem is an abstraction of motif finding, a common bioinformatics discovery task. The PTAS due to Li et al. is simple, and a preliminary implementation by Liang 2001 gave reasonable results in practice. However, the previously known bounds on its performance are useless when runtimes are actually manageable. Here, we present much sharper lower and upper bounds on the performance of this algorithm that partially explain why its behavior is so much better in practice than what was previously predicted in theory. We also give specific examples of instances of the problem for which the PTAS performs poorly in practice, and show that the asymptotic performance bound given in the original proof matches the behaviour of a simple variant of the algorithm on a particularly bad instance of the problem.