Packing to angles and sectors

Motivated by the widespread proliferation of wireless networks employing directional antennas, we study some capacitated covering problems arising in these networks. Geometrically, the area covered by a directional antenna with parameters α,r is a set of points with polar coordinates (r,) such that r r and α α + . Given a set of customers, their positions on the plane and their bandwidth demands, the capacitated covering problem considered here is to cover all the customers with the minimum number of directional antennas such that the demands of customers assigned to an antenna stays within a bound. We consider two settings of this capacitated cover problem arising in wireless networks. In the first setting where the antennas have variable angular range, we present an approximation algorithm with ratio 3. In the setting where the angular range of antennas is fixed, we improve this approximation ratio to 1.5. These results also apply for a related problem of bin packing with deadlines. In this problem we are are given a set of items, each with a weight, arrival time and deadline, and we want to pack each item into a bin after it arrives but before its deadline. The objective is to minimize the number of bins used. We present a 3-approximation algorithm for this problem, and 1.5-approximation algorithm for the special case when each difference between a deadline and the corresponding arrival time is the same.