# Information Security and Cryptography Research Group

## On the Parameterized Complexity of the Fixed Alphabet Shortest Common Supersequence and Longest Common Subsequence Problems

### Krzysztof Pietrzak

Journal of Computer and System Sciences, vol. 67, no. 1, pp. 757–771, Aug 2003.

We show that the Fixed Alphabet Shortest Common Supersequence (SCS) and the Fixed Alphabet Longest Common Subsequence (LCS) problems parameterized in the number of strings are W[1]-hard. Unless W[1]=FPT, this rules out the existence of algorithms with time complexity of O(f(k)n^c) for those problems. Here n is the size of the problem instance, c is constant, k is the number of strings and f is any function of k. The fixed alphabet version of the LCS problem is of particular interest considering the importance of sequence comparison (e.g. multiple sequence alignment) in the fixed length alphabet world of DNA and protein sequences.

## BibTeX Citation

@article{Pietrz03,
author       = {Krzysztof Pietrzak},
title        = {On the Parameterized Complexity of the Fixed Alphabet Shortest Common Supersequence and Longest Common Subsequence Problems},
journal      = {Journal of Computer and System Sciences},
pages        = 757--771,
number       = 1,
volume       = 67,
year         = 2003,
month        = 8,
}