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 #1135750. 	[Title] TREE CONTRACTIONS AND EVOLUTIONARY TREES [Abstract] An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of species. An agreement subtree of two evolutionary trees is an evolutionary tree which is also a topological subtree of the two given trees. We give an algorithm to determine the largest possible number of leaves in any agreement subtree of two trees T 1 and T 2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log 2 n) and is within a log n factor of optimal. For general d, this algorithm runs in O(nd 2 log d log 2 n) time or alternately in O(nd p d log 3 n) time. Please summarise all the given information.

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 #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.

Above are papers related to the following paper: [Paper_ID] Paper #1138043. 	[Title] Inserting the best known bounds for weighted bipar tite matching [11], with 1=2 p polynomial-time [Abstract] we apply the reduction to their two core children, the total sum of their matching weights becomes O(n), and if for each comparison of a spine node and a critical node we apply the reduction to the core child of the spine node, the total sum of their matching weights becomes O(n). With regards to the O( 2 ) comparisons of two critical nodes, their sum cannot exceed O( 2 n) in total weight. Thus, since we have a total of O(n) edges involved in the matchings, in time O(n), we can reduce the total sum of the matching weights to O( 2 n). Theorem 6.6 Let M : IR 1 ! IR be a monotone function bounding the time complexity UWBM. Moreover, let M satisfy that M (x) = x 1+" f(x), where " 0 is a constant, f (x) = O(x o(1) ), f is monotone, and for some constants b 1 ; b 2 , 8x; y b 1 : f(xy) b 2 f (x)f (y). Then, with = " p in time O(n 1+o(1) + M (n)). Proof: We spend O(n polylog n + time(UWBM(k 2 n))) on the matchings. So, by Theorem 5.10, we have that Comp-Core-Trees can be computed in time O(n polylog n + time(UWBM(k 2 n))). Applying Theo 4 = 16, we get Corollary 6.7 MAST is computable in time O(n 1:5 log n). [4] W.H.E. Day. Computational complexity of inferring phylogenies from dissimilarity matrices. Bulletin of Mathematical Biology, 49(4):461-467, 1987. [5] M. Farach, S. Kannan, and T. Warnow. A robust model for finding optimal evolutionary trees. Al-gorithmica, 1994. In press. See also STOC '93. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #573535. 	[Title] Fast and Simple Algorithms for Perfect Phylogeny and Triangulating Colored Graphs [Abstract] This paper presents an O((r n=m) m rnm) algorithm for determining whether a set of n species has a perfect phylogeny, where m is the number of characters used to describe a species and r is the maximum number of states that a character can be in. The perfect phylogeny algorithm leads to an O((2e=k) k e 2 k) algorithm for triangulating a k-colored graph having e edges. 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.

Paper #695284 has the title `The Hardness of Problems on Thin Colored Graphs  
` and the 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 
`.
Question: Which category should Paper #695284 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #695284 has the title `The Hardness of Problems on Thin Colored Graphs  
` and the 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 
`.
Question: Which category should Paper #695284 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