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.

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Above are papers related to the following paper: [Paper_ID] Paper #77112. 	[Title] A Six-Point Condition for Ordinal Matrices  keywords: additive, algorithm, evolution, ordinal, phylogeny [Abstract] Ordinal assertions in an evolutionary context are of the form "species s is more similar to species x than to species y" and can be deduced from a distance matrix M of interspecies dissimilarities (M [s; x] &lt; M [s; y]). Given species x and y, the ordinal binary character c xy of M is defined by c xy (s) = 1 if and only if M [s; x] &lt; M[s; y], for all species s. In this paper we present several results concerning the inference of evolutionary trees or phylogenies from ordinal assertions. In particular, we present A six-point condition that characterizes those distance matrices whose ordinal binary characters are pairwise compatible. This characterization is analogous to the four-point condition for additive matrices. An optimal O(n 2 ) algorithm, where n is the number of species, for recovering a phylogeny that realizes the ordinal binary characters of a distance matrix that satisfies the six-point condition. An NP-completeness result on determining if there is a phylogeny that realizes k or more of the ordinal binary characters of a given distance matrix. Please summarise all the given information.

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Paper #1138043 has the title `Inserting the best known bounds for weighted bipar tite matching [11], with 1=2 p polynomial-time
` and the 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. 
`.
Question: Which category should Paper #1138043 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #1138043 has the title `Inserting the best known bounds for weighted bipar tite matching [11], with 1=2 p polynomial-time
` and the 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. 
`.
Question: Which category should Paper #1138043 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