Service-constrained network design problems

Abstract

Several practical instances of network design problems often require the network to satisfy multiple constraints. In this paper, we focus on the following problem (and its variants): find a low-cost network, under one cost function, that services every node in the graph, under another cost function, (i.e., every node of the graph is within a prespecified distance from the network). This study has important applications to the problems of optical network design and the efficient maintenance of distributed databases.

We utilize the framework developed in [MR+95] to formulate these problems as bicriteria network design problems. We present the first known approximation algorithms for the class of service-constrained network design problems. We provide a (1, O($\tilde \Delta$ ln n))-approximation algorithm for the (Service cost, Total edge cost, Tree)-bicriteria problem (where $\tilde \Delta$ is the maximum service-degree of any node in the graph). We counterbalance this by showing that the problem does not have an (α, β)-approximation algorithm for any α ≥ 1 and β < ln n unless NP $\subseteq$ DTIME(n ^{log log n}). When both the objectives are evaluated under the same cost function we provide a (2(1 + ε), 2(1 + 1/ε))-approximation algorithm, for any ε >0. In the opposite direction we provide a hardness result showing that even in the restricted case where the two cost functions are the same the problem does not have an (α, β)-approximation algorithm for α = 1 and β < ln n unless NP $\subseteq$ DTIME(n ^{log log n}). We also consider a generalized Steiner forest version of the problem along with other variants involving diameter and bottleneck cost.

Research supported by the Department of Energy under Contract W-7405-ENG-36.

Research supported by DARPA contract N0014-92-J-1799 and NSF 92-12184 CCR.