Bicriteria Network Design Problems

Abstract

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a ¡subgraph from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria -the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented.First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same we present a "black box" parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms -the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.

Topology Control Problems under Symmetric and Asymmetric Power Thresholds

“…Our results for the diameter problem use the bicriteria approximation framework developed in [11] for dealing with computationally intractable optimization problems involving two objectives. We recall the relevant definitions and notation.…”

“…Mctdc is known to be NP-hard [11]. Bicriteria approximations for this problem have been presented in [2,9,11].…”

“…Bicriteria approximations for this problem have been presented in [2,9,11]. These results are used in Section 4.…”

“…For any fixed > 0, a (2 log 2 n , (1+ ) log 2 n )-approximation algorithm is presented in [11] for the Mctdc problem. Using this algorithm and setting < 1/n, we can obtain a (4 log 2 n , 2 log 2 n )-approximation algorithm for the Undir, Diameter, TotalP problem.…”

“…We can also obtain an approximation algorithm for the Steiner version of the Undir, Diameter, TotalP problem where only a specified subset of the nodes (called the terminals) need to be connected together into a graph of bounded diameter. Letting η denote the number of terminals, reference [11] presents an (O(log η), O(log η))-approximation algorithm for the Steiner version of the Mctdc problem. Using this approximation algorithm in Step 2 of Figure 1, we obtain an (O(log η), O(log η))-approximation algorithm for the Steiner version of the Undir, Diameter, TotalP problem.…”