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Consequences of Faster Alignment of Sequences
"... Abstract. The Local Alignment problem is a classical problem with applications in biology. Given two input strings and a scoring function on pairs of letters, one is asked to find the substrings of the two input strings that are most similar under the scoring function. The best algorithms for Local ..."
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Abstract. The Local Alignment problem is a classical problem with applications in biology. Given two input strings and a scoring function on pairs of letters, one is asked to find the substrings of the two input strings that are most similar under the scoring function. The best algorithms for Local Alignment run in time that is roughly quadratic in the string length. It is a big open problem whether substantially subquadratic algorithms exist. In this paper we show that for all ε> 0, an O(n2−ε) time algorithm for Local Alignment on strings of length n would imply breakthroughs on three longstanding open problems: it would imply that for some δ> 0, 3SUM on n numbers is in O(n2−δ) time, CNFSAT on n variables is in O((2 − δ)n) time, and Max Weight 4Clique is in O(n4−δ) time. Our result for CNFSAT also applies to the easier problem of finding the longest common substring of binary strings with don’t cares. We also give strong conditional lower bounds for the more general Multiple Local Alignment problem on k strings, under both kwise and SP scoring, and for other string similarity problems such as Global Alignment with gap penalties and normalized Longest Common Subsequence. 1
Parameterized Complexity and Biopolymer Sequence Comparison
, 2007
"... The paper surveys parameterized algorithms and complexities for computational tasks on biopolymer sequences, including the problems of longest common subsequence, shortest common supersequence, pairwise sequence alignment, multiple sequencing alignment, structure–sequence alignment and structure–str ..."
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The paper surveys parameterized algorithms and complexities for computational tasks on biopolymer sequences, including the problems of longest common subsequence, shortest common supersequence, pairwise sequence alignment, multiple sequencing alignment, structure–sequence alignment and structure–structure alignment. Algorithm techniques, built on the structuralunit level as well as on the residue level, are discussed.
W hardness under linear FPTreductions: structural properties and further applications
 in Proceedings of the The Eleventh International Computing and Combinatorics Conference (COCOON’05
, 2005
"... Abstract. The notion of linear fptreductions has been recently used to derive strong computational lower bounds for wellknown NPhard problems. In this paper, we formally investigate the notions of W [t]hardness and W [t]completeness under the linear fptreduction, and study structural propertie ..."
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Abstract. The notion of linear fptreductions has been recently used to derive strong computational lower bounds for wellknown NPhard problems. In this paper, we formally investigate the notions of W [t]hardness and W [t]completeness under the linear fptreduction, and study structural properties of the corresponding complexity classes. Additional complexity lower bounds on important computational problems are also established. 1
Applications of PARAMETERIZED Computation in . . .
, 2005
"... This paper first gives an introduction to parameterized computation and complexity theory, a new subfield in theoretical computer science. Then it presents a summary of its applications to addressing some important NPhard problems in computational biology. Specifically, we can design efficient para ..."
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This paper first gives an introduction to parameterized computation and complexity theory, a new subfield in theoretical computer science. Then it presents a summary of its applications to addressing some important NPhard problems in computational biology. Specifically, we can design efficient parameterized algorithms and also drive computational lower bounds for the parameterized algorithms and approximation algorithms of computational biological problems.
Structure of PolynomialTime Approximation
, 2009
"... Approximation schemes are commonly classified as being either a polynomialtime approximation scheme (ptas) or a fully polynomialtime approximation scheme (fptas). To properly differentiate between approximation schemes for concrete problems, several subclasses have been identified: (optimum)asymp ..."
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Approximation schemes are commonly classified as being either a polynomialtime approximation scheme (ptas) or a fully polynomialtime approximation scheme (fptas). To properly differentiate between approximation schemes for concrete problems, several subclasses have been identified: (optimum)asymptotic schemes (ptas∞ , fptas∞), efficient schemes (eptas), and sizeasymptotic schemes. We explore the structure of these subclasses, their mutual relationships, and their connection to the classic approximation classes. We prove that several of the classes are in fact equivalent. Furthermore, we prove the equivalence of eptas to socalled convergent polynomialtime approximation schemes. The results are used to refine the hierarchy of polynomialtime approximation schemes considerably and demonstrate the central position of eptas among approximation schemes. We also present two ways to bridge the hardness gap between asymptotic approximation schemes and classic approximation schemes. First, using notions from fixedparameter complexity theory, we provide new characterizations of when problems have a ptas or fptas. Simultaneously, we prove that a large class of problems (including all MAXSNPcomplete