No abstract

In this paper, we consider different aspects of robust 1‐median problems on a tree network with uncertain or dynamically changing edge lengths and vertex weights which can also take negative values. The dynamic nature of a parameter is modeled by a linear function of time. A linear algorithm is designed for the absolute dynamic robust 1‐median problem on a tree. The dynamic robust deviation 1‐median problem on a tree with n vertices is solved in O(n2 α(n) log n) time, where α(n) is the inverse Ackermann function. Examples show that both problems do not possess the vertex optimality property. The uncertainty is modeled by given intervals, in which each parameter can take a value randomly. The absolute robust 1‐median problem with interval data, where vertex weights might also be negative, can be solved in linear time. The corresponding deviation problem can be solved in O(n2) time. © 2001 John Wiley & Sons, Inc.

No abstract

The obnoxious center problem in a graph G asks for a location on an edge of the graph such that the minimum weighted distance from this point to a vertex of the graph is as large as possible. We derive algorithms with linear running time for the cases when G is a path or a star, thus improving previous results of Tamir [SIAM J. Discrete Math, 1 (1988), pp. 377-396]. For subdivided stars we present an algorithm of running time O(n log n). For general trees, we improve an algorithm of Tamir [SIAM J. Discrete Math, 1 (1988), pp. 377-396] by a factor of log n. Moreover, a linear algorithm for the unweighted center problem on an arbitrary tree with neutral and obnoxious vertices is described.

We consider the robust 1-center problem on trees with uncertainty in vertex weights and edge lengths. The weights of the vertices and the lengths of the edges can take any value in prespecified intervals with unknown distribution. We show that this problem can be solved in O(n 3 log n) time thus improving on Averbakh and Berman's algorithm with time complexity O(n 6 ). For the case when the vertices of the tree have weights equal to 1 we show that the robust 1-center problem can be solved in O(n log n) time, again improving on Averbakh and Berman's time complexity of O(n 2 log n).

No abstract