Approximating the Minimum Number of Maximum Power Users in Ad Hoc Networks

Topology control is the problem of assigning transmission power values to the nodes of an ad hoc network so that the induced graph satisfies some specified property. The most fundamental such property is that the network/graph be connected. For connectivity, prior work on topology control gave a polynomial time algorithm for minimizing the maximum power assigned to any node (such that the induced graph is connected). In this paper we study the problem of minimizing the number of maximum power nodes. After establishing that this minimization problem is NP-Complete, we focus on approximation algorithms for graphs with symmetric power thresholds. We first show that the problem is reducible in an approximation preserving manner to the problem of assigning power values so that the sum of the powers is minimized. Using known results for that problem, this provides a family of approximation algorithms for the problem of minimizing the number of maximum power nodes with approximation ratios of 5/3 + for every > 0. Unfortunately, these algorithms, based on solving large linear programming problems are not practical. The main result of this paper is a practical algorithm having a 5/3 (exactly) approximation ratio. In addition, we present experimental results on randomly generated networks. Finally, based on the reduction to minimizing the total power problem, we outline some additional results for minimizing the number of maximum power users, both for graph properties other than connectivity and for graphs with asymmetric power thresholds.