Budget constrained minimum cost connected medians

Several practical instances of network design and location theory problems require the network to satisfy multiple constraints. In this paper, we study a graph-theoretic problem that aims to simultaneously address a network design task and a location-theoretic constraint. The Budget Constrained Connected Median Problem is the following: We are given an undirected graph G=(V,E) with two different edge-weight functions c (modeling the construction or communication cost) and d (modeling the service distance), and a bound B on the total service distance. The goal is to find a subtree T of G with minimum c-cost c(T) subject to the constraint that the sum ∑_vVTdist_d(v,T) of the service distances of all the remaining nodes vVT does not exceed the specified budget B. Here, the service distance dist_d(v,T) denotes the shortest path distance of v to a vertex in T with respect to d. This problem has applications in optical network design and the efficient maintenance of distributed databases.

We formulate this problem as a bicriteria network design problem, and present bicriteria approximation algorithms. We also prove lower bounds on the approximability of the problem which demonstrate that our performance ratios are close to best possible.