Background The history of gene families-which are equivalent to event-labeled gene trees-can to some extent be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, par...
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Background The history of gene families-which are equivalent to event-labeled gene trees-can to some extent be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes. The question then arises as whether inferred event-labeled gene trees are "biologically feasible" which is the case if one can find a species tree with which the gene tree can be reconciled in a time-consistent way. Results In this contribution, we consider event-labeled gene trees that contain speciations, duplications as well as horizontal gene transfer (HGT) and we assume that the species tree is unknown. Although many problems become NP-hard as soon as HGT and time-consistency are involved, we show, in contrast, that the problem of finding a time-consistent species tree for a given event-labeled gene can be solved in polynomial-time. We provide a cubic-timealgorithm to decide whether a "time-consistent" species tree for a given event-labeled gene tree exists and, in the affirmative case, to construct the species tree within the same time-complexity.
We present a new manifestation of Godel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension ...
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We present a new manifestation of Godel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension of Peano arithmetic (PA) by a mathematically meaningful axiom scheme that consists of Sigma(0)(2)-sentences. These sentences assert that each computably enumerable Sigma(0)(2)-definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomialtimealgorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Godel's second incompleteness theorem. In contrast, Godel's theorem is not needed for typical examples of Pi(0)(2)-independence (such as the Paris-Harrington principle), since computational strength provides an extensional invariant on the level of Pi(0)(2)-sentences.
Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. First, we introduce a common generalization of th...
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Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. First, we introduce a common generalization of these two concepts for bipartite plane graphs, and then we extend it further to the notion of source-sink pairs of subsets of nodes in a general (not necessarily planar) directed graph. The main result is a min-max formula for the maximum weight of a source-sink pair. The proof is based on the recognition that the convex hull of source-sink pairs can be obtained as the projection of a network tension polyhedron. The construction makes it possible to apply any standard cheapest network flow algorithm to compute both a maximum weight source-sink pair and a minimizer of the dual optimization problem formulated in the min-max theorem. As a consequence, our approach gives rise to the first purely combinatorial, strongly polynomialalgorithm to compute a largest (or even a maximum weight) Fries-set of a perfectly matchable plane bipartite graph and an optimal solution to the dual minimization problem. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are expl...
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Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.
The following class of matching problems is considered. The vertices of a complete undirected graph are indexed 1,⋯,n1,⋯,n1, \cdots,n, where <span class="MathJax" id="MathJax-Element-5-Frame" ta...
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We introduce the problem of adapting a stable matching to forced and forbidden pairs. Specifically, given a stable matching M1, a set Q of forced pairs, and a set P of forbidden pairs, we want to find a stable matchin...
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ISBN:
(纸本)9781450394321
We introduce the problem of adapting a stable matching to forced and forbidden pairs. Specifically, given a stable matching M1, a set Q of forced pairs, and a set P of forbidden pairs, we want to find a stable matching that includes all pairs from Q, no pair from P, and that is as close as possible to M1. We study this problem in four classical stable matching settings: Stable Roommates (with Ties) and Stable Marriage (with Ties).As our main contribution, we employ the theory of rotations for Stable Roommates to develop a polynomial-time algorithm for adapting Stable Roommates matchings to forced pairs. In contrast to this, we show that the same problem for forbidden pairs is NP-hard. However, our polynomial-time algorithm for the case of only forced pairs can be extended to a fixed-parameter tractable algorithm with respect to the number of forbidden pairs when both forced and forbidden pairs are present. Moreover, we also study the setting where preferences contain ties. Here, depending on the chosen stability criterion, we show either that our algorithmic results can be extended or that formerly tractable problems become intractable.
Given an undirected graph G = (V, E) and a directed graph D = (V, A), the master/slave matching problem is to find a matching of maximum cardinality in G such that for each arc (u, v) ε A with u being matched, v is a...
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ISBN:
(纸本)9781921770098
Given an undirected graph G = (V, E) and a directed graph D = (V, A), the master/slave matching problem is to find a matching of maximum cardinality in G such that for each arc (u, v) ε A with u being matched, v is also matched. This problem is known to be NP-hard in general, but polynomially solvable in a special case where the maximum size of a connected component of D is at most *** paper investigates the master/slave matching problem in terms of delta-matroids, which is a generalization of matroids. We first observe that the above polynomially solvable constraint can be interpreted as delta-matroids. We then introduce a new class of matching problem with delta-matroid constraints, which can be solved in polynomialtime. In addition, we discuss our problem with additional constraints such as capacity constraints.
The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which ...
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ISBN:
(纸本)9780898716054
The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e. finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs.
The location problem is one kind of special type optimized problems .The Min-max weighted distance problem is a new class of location problem, its decision problem is a NP- Complete problem. In this paper, some approx...
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The location problem is one kind of special type optimized problems .The Min-max weighted distance problem is a new class of location problem, its decision problem is a NP- Complete problem. In this paper, some approximate algorithms are designed, designs a genetic algorithm by some properties of the problem, and gives the design and selection method of crossover operator, mutation operator and reproduction operator.
The location problem is one kind of special type optimized problems .The Min-max weighted distance problem is a new class of location problem, its decision problem is a NP-Complete problem. In this paper, some approxi...
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The location problem is one kind of special type optimized problems .The Min-max weighted distance problem is a new class of location problem, its decision problem is a NP-Complete problem. In this paper, some approximate algorithms are designed, designs a genetic algorithm by some properties of the problem, and gives the design and selection method of crossover operator, mutation operator and reproduction operator.
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