Some intersection theorems for ordered sets and graphs
Journal of Combinatorial Theory Series A
On the efficiency of polynomial time approximation schemes
Information Processing Letters
On the closest string and substring problems
Journal of the ACM (JACM)
On the Parameterized Intractability of CLOSEST SUBSTRINGsize and Related Problems
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
On the complexity of finding common approximate substrings
Theoretical Computer Science
Tight Lower Bounds for Certain Parameterized NP-Hard Problems
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Constraint solving via fractional edge covers
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
An improved lower bound on approximation algorithms for the Closest Substring problem
Information Processing Letters
More efficient algorithms for closest string and substring problems
RECOMB'08 Proceedings of the 12th annual international conference on Research in computational molecular biology
Algorithms for multiterminal cuts
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
A three-string approach to the closest string problem
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
More Efficient Algorithms for Closest String and Substring Problems
SIAM Journal on Computing
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In the CLOSEST SUBSTRING problem k strings s1, . . . sk are given, and the task is to find a string s of length L such that each string si has a consecutive substring of length L whose distance is at most d from s. The problem is motivated by applications in computational biology. We present two algorithms that can be efficient for small fixed values of d and k: for some functions f and g, the algorithms have running time f(d)· n^o^{(\log d)}and g(d,k)·n^o ^{(\log log)},respectively. The second algorithm is based on connections with the extremal combinatorics of hypergraphs. The CLOSEST SUBSTRING problem is also investigated from the parameterized complexity point of view. Answering an open question from [6, 7, 11, 12], we show that the problem is W[1]- hard even if both d and k are parameters. It follows as a consequence of this hardness result that our algorithms are optimal in the sense that the exponent of n in the running time cannot be improved to o(logd) or to o(log log k) (modulo some complexity0-theoretic assumptions). Another consequence is that the running time n^o ^{(1/\varepsilon^4)}of the approximation scheme for CLOSEST SUBSTRING presented in [13] cannot be improved to f (\varepsilon) · {n^c}, i.e., the \varepsilon has to appear in the exponent of n.