The closest string and substring problems find applications in PCR primer design, genetic probe design, motif finding, and antisense drug design. For their importance, the two problems have been extensively studied recently in computational biology. Unfortunately both problems are NP-complete. Researchers have developed both fixed-parameter algorithms and approximation algorithms for the two problems. In terms of fixed-parameter, when the radius d is the parameter, the best-known fixedparameter algorithm for closest string has time complexity O(ndd+1), which is still superpolynomial even if d = O(log n). In this paper we provide an O � n|Σ | O(d) � algorithm where Σ is the alphabet. This gives a polynomial time algorithm when d = O(log n) and Σ has constant size. Using the same technique, we additionally provide a more efficient subexponential time algorithm for the closest substring problem. In terms of approximation, both closest string and closest substring problems admit polynomial time approximation schemes (PTAS). The best known time complexity of the PTAS is O(n O(ɛ−2 log 1 ɛ)). In this paper we present a PTAS with time complexity O(n O(ɛ−2)). At last, we prove that a restricted version of the closest substring has the same parameterized complexity as closest substring, answering an open question in the literature
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