Given a graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider the following fundamental problem in wireless network design. Given a graph G = (V, E) with edge costs and degree bounds {r(v) : v ∈ V }, the Minimum-Power Edge-Multi-Cover (MPEMC) problem is to find a minimum-power subgraph J of G such that the degree of every node v in J is at least r(v). We give two approximation algorithms for MPEMC, with ratios O(log k) and k + 1/2, where k = maxv∈V r(v) is the maximum degree bound. This improves the previous ratios O(log n) and k + 1, and implies ratios O(log k) for the Minimum-Power k-Outconnected Subgraph and O log k log n n−k for the Minimum-Power k-Connected Subgraph problems; the latter is the currently best known ratio for the min-cost version of the problem.
Introduction
Motivation and problems consideredWireless networks are studied extensively due to their wide applications. The power consumption of a station determines its transmission range, and thus also the stations it can send messages to; the power typically increases at least quadratically in the transmission range. Assigning power levels to the stations (nodes) determines the resulting communication network. Conversely, given a communication network, the power required at v only depends on the farthest node reached directly by v. This is in contrast with wired networks, in which every pair of stations that communicate directly incurs a cost. An important network property is fault-tolerance, which is often measured by minimum degree or node-connectivity of the network. Node-connectivity is much more central here than edge-connectivity, as it models stations failures. Such power minimization problems were vastly studied; see for example [1,2,5,8,9] and the references therein for a small sample of papers in this area. The first problem we consider is finding a low power network with specified lower degree bounds. The second problem is the Min-Power k-Connected Subgraph problem. We give approximation algorithms for these problems, improving the previously best known ratios.