Showing papers on "Betweenness centrality published in 1998"

Relations related to betweenness : their structure and automorphisms

Abstract: Preparation Semilinear order relations Abstract chain sets General betweenness relations Abstract direction sets Applications and commentary References.

A Geometric Approach to Betweenness

Abstract: An input to the betweenness problem contains m constraints over n real variables (points). Each constraint consists of three points, where one of the points is specified to lie inside the interval defined by the other two. The order of the other two points (i.e., which one is the largest and which one is the smallest) is not specified. This problem comes up in questions related to physical mapping in molecular biology. In 1979, Opatrny showed that the problem of deciding whether the n points can be totally ordered while satisfying the m betweenness constraints is NP-complete [SIAM J. Comput., 8 (1979), pp. 111--114]. Furthermore, the problem is MAX SNP complete, and for every $\alpha> 47/48$ finding a total order that satisfies at least $\alpha$ of the m constraints is NP-hard (even if all the constraints are satisfiable). It is easy to find an ordering of the points that satisfies 1/3 of the m constraints (e.g., by choosing the ordering at random). This paper presents a polynomial time algorithm that either determines that there is no feasible solution or finds a total order that satisfies at least 1/2 of the m constraints. The algorithm translates the problem into a set of quadratic inequalities and solves a semidefinite relaxation of them in ${\cal R}^n. The n solution points are then projected on a random line through the origin. The claimed performance guarantee is shown using simple geometric properties of the semidefinite programming (SDP) solution.

Consecutive Ones and a Betweenness Problem in Computational Biology

Abstract: In this paper we consider a variant of the betweenness prob- lem occurring in computational biology. We present a new polyhedral approach which incorporates the solution of consecutive ones problems and show that it supersedes an earlier one. A particular feature of this new branch-and-cut algorithm is that it is not based on an explicit integer programming formulation of the problem and makes use of automatically generated facet-defining inequalities.

Relations related to betweenness

Abstract: In the last chapter we defined linear betweenness relations, circular (or cyclic) orders and separation relations from a linear order and studied their groups of automorphisms. The automorphism group of a linear order has already been studied in detail in Chapter 9. That of a circular order can be understood best in terms of the linear order obtained by deleting a point. The groups of automorphisms of a linear betweenness relation and of a separation relation are simply the groups of order-preserving or order-reversing transformations on a linearly ordered and cyclically ordered set respectively. In this chapter we introduce four more relations related to betweenness which will lead us to the classification of primitive Jordan groups that have primitive Jordan sets. Everything discussed in this chapter has been discussed in greater detail and rigour in Adeleke & Neumann (1996c). The arguments used in this chapter are very geometric and we encourage the reader to draw pictures. Note however that our semilinear orders grow upwards rather than downwards, contrary to the convention followed in Adeleke & Neumann (1996c).

An Approach to Modelling Directional and Betweenness Data

Abstract: A model is introduced to analyze and represent spatially either directly obtained or derived judgments expressing directions in space or betwennness relations among objects. For the two-dimensinal case the construction rules tofind a spatial representation of the object points are given in detail. A small example illustrates the procedure. Consistency tests are reported which allow a differentiated analysis of this kind of cognitive structure. The three-dimensional case is discussed briefly. Finally, it is illustrated how ratings, rankings, and co-occurrence informations are transformed to betweenness data.