Optimal algorithms for computing edge weights in planar split networks

In phylogenetics, biologists commonly compute split networks when trying to better understand evolutionary data. These graph-theoretical structures represent collections of weighted bipartitions or splits of a finite set, and provide a means to display conflicting evolutionary signals. The weights associated to the splits are used to scale the edges in the network and are often computed using some distance matrix associated with the data. In this paper we present optimal polynomial time algorithms for three basic problems that arise in this context when computing split weights for planar split-networks. These generalize algorithms that have been developed for special classes of split networks (namely, trees and outer-labeled planar networks). As part of our analysis, we also derive a Crofton formula for full flat split systems, structures that naturally arise when constructing planar split-networks.     

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.     

A parallel algorithm for generating multiple ordering spanning trees in undirected weighted graphs

1.IntroductionLetG=(V,E,W)beaconnected,weightedandundirectedgraph,VeEE,w(e)(相似文献   

Consistency of the instrumental weighted variables

A robust version of method of Instrumental Variables accommodating the idea of an implicit weighting the residuals is proposed and its properties studied. Firstly, it is shown that all solutions of the corresponding normal equations are bounded in probability. Then the weak consistency of them is proved. The algorithm, evaluating the estimate, is described and results of small MC study discussed. Research was supported by grant of GA ČR number 402/06/0408.     

On the generating function for consecutively weighted permutations

We show that the analytic continuation of the exponential generating function associated to consecutive weighted pattern enumeration of permutations only has poles and no essential singularities. The proof uses the connection between permutation enumeration and functional analysis, and as well as the Laurent expansion of the associated resolvent. As a consequence, we give a partial answer to a question of Elizalde and Noy: when is the multiplicative inverse of the exponential generating function for the number permutations avoiding a single pattern an entire function? Our work implies that it is enough to verify that this function has no zeros to conclude that the inverse function is entire.     

Analysis of split weighted least-squares procedures for pseudo-hyperbolic equations

In this paper, we introduce two novel split weighted least-squares finite element procedures for pseudo-hyperbolic equations arising in the modelling of nerve conduction process. By selecting the weighted least-squares functional properly, each procedure can be split into two independent symmetric positive definite sub-procedures. One of sub-procedures is for the primitive unknown variable, which is the same as the standard Galerkin finite element procedure and the other is for the introduced flux variable. Optimal order error estimates are developed and the numerical example is given to show the efficiency of the introduced schemes.     

Linear programming based meta-heuristics for the weighted maximal planar graph

The weighted maximal planar graph (WMPG) is practically important in the laying out of facilities in modern manufacturing environments. Given a weighted complete graph, the WMPG seeks to find a sub-graph such that it is planar—it can be embedded on the plane without any arcs intersecting, and it is maximal—no additional arc can be added to the sub-graph without destroying its planarity, and it also has the maximal sum of arc weights. In this paper, an integer linear programming (ILP) model is newly introduced for the problem. Two meta-heuristics are then derived from the ILP relaxation. The first meta-heuristic considers all variables with fractional values greater than half in the ILP relaxation to build an initial sub-graph from which a planar sub-graph is extracted using greedy random adaptive search procedure (GRASP) and augmented by triangulation of faces. The second meta-heuristic considers only arcs with integer values in the ILP relaxation. The remaining arcs are then sorted in descending order of their weights, for selection and insertion with a planarity testing procedure, to obtain a feasible solution using GRASP. Computational results are reported on a set of 100 test instances of size varying from 20 to 100 facilities. The computational results demonstrate the tightness of the new upper bound when compared to the classical one as well as the good performance of the proposed metaheuristics when compared to the best-known procedures in the literature in terms of solution quality and computational requirement. Finally, the paper presents a successful integration of GRASP with classical optimisation approaches and should be attempted for other optimisation problems.     

Cooperative Cross-Entropy method for generating entangled networks

Survivability is rapidly becoming an important criterion in network design and planning. This is due to our increased dependence on ever more complex communication networks. Another important criterion which plays a central role in design and planning decisions is cost. As a result, network planners tend to design sparse networks to minimise cost. There is a class of networks known as entangled networks which seems to satisfy both criteria of survivability and sparseness. In this paper, we demonstrate how the Cross-Entropy method may be utilised to generate entangled networks. We also propose a cooperative optimisation approach to further improve the generation of an optimal entangled network.     

A new algorithm for generating derangements

A new algorithm for generating derangements based on a well known permutation generation method is presented and analysed. The algorithm is shown to be superior in storage and time requirements to the existing method.This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grant NSERC-A3336.     

A simplified loop-free algorithm for generating permutations

A simple loop-free algorithm for generation of all permutations of a set of elements is presented and its validity is proved. It is a simplification of Ehrlich's loop-free version of Johnson and Trotter's algorithm. Each permutation is generated by exchanging two adjacent elements of the preceding permutation. A very simple data structure obviates the need for looping during the generation of each successive permutation.     

A near-linear algorithm for the planar segment-center problem

LetP be a set ofn points in the plane and lete be a segment of fixed length. The segment-center problem is to find a placement ofe (allowing translation and rotation) which minimizes the maximum euclidean distance frome to the points ofP. We present an algorithm that solves the problem in timeO(n ^1+ε), for any ε>0, improving the previous solution of Agarwalet al. [3] by nearly a factor ofO(n). Work on this paper by the second author has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Practical strategies for generating rank-1 split cuts in mixed-integer linear programming

In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of these variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. An extensive computational evaluation of these cuts points to the following two conclusions. The first is that our rank-1 cuts improve significantly on existing split cut generators (Gomory cuts from single tableau rows, MIR, Reduce-and-Split, Lift-and-Project, Flow and Knapsack cover): on MIPLIB instances, these generators close 24% of the integrality gap on average; adding our cuts yields an additional 5%. The second conclusion is that, when incorporated in a Branch-and-Cut framework, these new cuts can improve computing time on difficult instances.     

An algorithm for surface reconstruction from planar contours using smoothing splines

This paper presents a fast algorithm for constructing a smooth three-dimensional surface over a set of cross-sectional contours. We assume that these sections are perpendicular to the z-axis and first consider the case that the surface can be represented in cylindrical coordinates. An approximation is then determined for r(θ, z) by using tensor product splines which satisfy certain boundary constraints. The algorithm is an extension of an existing semi-automatic surface fitting algorithm. The knots of the spline are chosen automatically but a single parameter is expected to control the tradeoff between closeness of fit and smoothness of fit.Both open and closed surfaces can be represented. In particular we demonstrate the use of a non-linear transformation for obtaining smooth closed surfaces.The algorithm can easily be extended to the reconstruction of surfaces which cannot be represented in cylindrical coordinates. A number of applications are also briefly discussed such as the calculation of volumes and the intersection with other surfaces. We have applied the method in practice to obtain a 3-D integrated image of the cerebral blood vessels and CT view of tumor for stereotactic neurosurgery.     

加权网络结构分析

网络研究已经成为机器学习领域中的热点问题之一,近年来发展起来的随机块模型是通过建模生成网络的一种方法．本文对随机块模型加以推广,建立加权的随机块模型,在求解过程中,采用一种可以广泛的用于求解混合模型的变分EM算法．最后通过数据模拟,证明了此方法的可行性．     

Willmore flow of planar networks

Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.     

Multigrid algorithm from cyclic reduction for Markovian queueing networks

A multigrid method based on cyclic reduction strategy is proposed to solve huge, nonsymmetric singular linear systems arising from Markovian queueing networks. A simple way to construct the matrix-dependent prolongation and restriction operators is presented in this paper. Numerical results for multiple queues are given to illustrate the efficiency and robustness of our methods.     

A left-first search algorithm for planar graphs

We give anO(|V(G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovič. The edges of any maximal bipartite plane graphG with outer facebwb′w′ can be colored by two colors such that the color classes form spanning trees ofG−b andG−b′, respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear-time algorithm for constructing bipolar orientations of 2-connected plane graphs. The research of H. de Fraysseix and P. O. de Mendez was supported by ESPRIT Basic Research Action No. 7141 (ALCOM II). J. Pach's research was supported by NSF Grant CCR-91-22103, OTKA-4269, and ALCOM II.     

A triangulation algorithm for arbitrary planar domains

For finite element problems which involve irregular regions it is useful to have an automatic scheme capable of forming consistent, high-quality triangulations economically and efficiently. Existing computer algorithms tend either to be lacking in flexibility or inordinately time consuming. This algorithm was developed for the FEHPOL system and has significant advantages over previous methods. A flow chart and computer listing are included.