Above are papers related to the following paper: [Paper_ID] Paper #132821. 	[Title] The Parameterized Complexity of Sequence Alignment and Consensus [Abstract] The Longest common subsequence problem is examined from the point of view of parameterized computational complexity. There are several different ways in which parameters enter the problem, such as the number of sequences to be analyzed, the length of the common subsequence, and the size of the alphabet. Lower bounds on the complexity of this basic problem imply lower bounds on a number of other sequence alignment and consensus problems. At issue in the theory of parameterized complexity is whether a problem which takes input (x; k) can be solved in time f (k) n ff where ff is independent of k (termed fixed-parameter tractability). It can be argued that this is the appropriate asymptotic model of feasible computability for problems for which a small range of parameter values covers important applications | a situation which certainly holds for many problems in biological sequence analysis. Our main results show that: (1) The Longest Common Subsequence (LCS) parameterized by the number of sequences to be analyzed is hard for W [t] for all t. (2) The LCS problem problem, parameterized by the length of the common subsequence, belongs to W [P ] and is hard for W [2]. (3) The LCS problem parameterized both by the number of sequences and the length of the common subsequence, is complete for W [1]. All of the above results are obtained for unrestricted alphabet sizes. For alphabets of a fixed size, problems (2) and (3) are fixed-parameter tractable. We conjecture that (1) remains hard. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #66751. 	[Title] A Faster Algorithm for the Perfect Phylogeny Problem when the number of Characters is Fixed TR94-05 [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #695284. 	[Title] The Hardness of Problems on Thin Colored Graphs [Abstract] In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded treewidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"-instances. For all of these problems with the exceptions of feasible register assignment and module allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 Z + . We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #693143. 	[Title] A Fast Algorithm for the Computation and Enumeration of Perfect Phylogenies [Abstract] The Perfect Phylogeny Problem is a classical problem in computational evolutionary biology, in which a set of species/taxa is described by a set of qualitative characters. In recent years, the problem has been shown to be NP-Complete in general, while the different fixed parameter versions can each be solved in polynomial time. In particular, Agarwala and Fernandez-Baca have developed an O(2 3r (nk 3 +k 4 )) algorithm for the perfect phylogeny problem for n species defined by k r-state characters. Since commonly the character data is drawn from alignments of molecular sequences, k is the length of the sequences and can thus be very large (in the hundreds or thousands). Thus, it is imperative to develop algorithms which run efficiently for large values of k. In this paper we make additional observations about the structure of the problem and produce an algorithm for the problem that runs in time O(2 2r k 2 n). We also show how it is possible to efficiently build a structure that implicitly represents the set of all perfect phylogenies, and to randomly sample from that set. Please summarise all the given information.

Paper #132821 has the title `The Parameterized Complexity of Sequence Alignment and Consensus  
` and the abstract `The Longest common subsequence problem is examined from the point of view of parameterized computational complexity. There are several different ways in which parameters enter the problem, such as the number of sequences to be analyzed, the length of the common subsequence, and the size of the alphabet. Lower bounds on the complexity of this basic problem imply lower bounds on a number of other sequence alignment and consensus problems. At issue in the theory of parameterized complexity is whether a problem which takes input (x; k) can be solved in time f (k) n ff where ff is independent of k (termed fixed-parameter tractability). It can be argued that this is the appropriate asymptotic model of feasible computability for problems for which a small range of parameter values covers important applications | a situation which certainly holds for many problems in biological sequence analysis. Our main results show that: (1) The Longest Common Subsequence (LCS) parameterized by the number of sequences to be analyzed is hard for W [t] for all t. (2) The LCS problem problem, parameterized by the length of the common subsequence, belongs to W [P ] and is hard for W [2]. (3) The LCS problem parameterized both by the number of sequences and the length of the common subsequence, is complete for W [1]. All of the above results are obtained for unrestricted alphabet sizes. For alphabets of a fixed size, problems (2) and (3) are fixed-parameter tractable. We conjecture that (1) remains hard. 
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Question: Which category should Paper #132821 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #132821 has the title `The Parameterized Complexity of Sequence Alignment and Consensus  
` and the abstract `The Longest common subsequence problem is examined from the point of view of parameterized computational complexity. There are several different ways in which parameters enter the problem, such as the number of sequences to be analyzed, the length of the common subsequence, and the size of the alphabet. Lower bounds on the complexity of this basic problem imply lower bounds on a number of other sequence alignment and consensus problems. At issue in the theory of parameterized complexity is whether a problem which takes input (x; k) can be solved in time f (k) n ff where ff is independent of k (termed fixed-parameter tractability). It can be argued that this is the appropriate asymptotic model of feasible computability for problems for which a small range of parameter values covers important applications | a situation which certainly holds for many problems in biological sequence analysis. Our main results show that: (1) The Longest Common Subsequence (LCS) parameterized by the number of sequences to be analyzed is hard for W [t] for all t. (2) The LCS problem problem, parameterized by the length of the common subsequence, belongs to W [P ] and is hard for W [2]. (3) The LCS problem parameterized both by the number of sequences and the length of the common subsequence, is complete for W [1]. All of the above results are obtained for unrestricted alphabet sizes. For alphabets of a fixed size, problems (2) and (3) are fixed-parameter tractable. We conjecture that (1) remains hard. 
`.
Question: Which category should Paper #132821 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Theory