On the complexity of finding common approximate substrings

Problems associated with #nding strings that are within a speci#ed Hamming distance of a given set of strings occur in several disciplines. In this paper, we use techniques from parameterized complexity to assess non-polynomial time algorithmic options and complexity for the COMMON APPROXIMATE SUBSTRING (CAS) problem. Our analyses indicate under which parameter restrictions useful algorithms are possible, and include both class membership and parameterized reductions to prove class hardness. In order to achieve #xed-parameter tractability, either a #xed string length or both a #xed size alphabet and #xed substring length are su7cient. Fixing either the string length or the alphabet size and Hamming distance is shown to be necessary, unless W [1] = FPT . An assortment of parameterized class membership and hardness results cover all other parameterized variants, showing in particular the e<ect of #xing the number of strings. c © 2003 Elsevier B.V. All rights reserved.