Suppose we are given a set of objects that cover a region and a duration associated with each object. Viewing the objects as jobs, can we schedule their beginning times to maximize the length of time that the original region remains covered? We call this problem the SENSOR COVER PROBLEM. It arises in the context of covering a region with sensors. For example, suppose you wish to monitor activity along a fence (interval) by sensors placed at various fixed locations. Each sensor has a range (also an interval) and limited battery life. The problem is then to schedule when to turn on the sensors so that the fence is fully monitored for as long as possible. This one-dimensional problem involves intervals on the real line. Associating a duration to each yields a set of rectangles in space and time, each specified by a pair of fixed horizontal endpoints and a height. The objective is to assign a bottom position to each rectangle (by moving them up or down) so as to maximize the height at which the spanning interval is fully covered. We call this one-dimensional problem RESTRICTED STRIP COVERING. If we replace the covering constraint by a packing constraint (rectangles may not overlap, and the goal is to minimize the highest point covered), then the problem becomes identical to DYNAMIC STORAGE ALLOCATION, a well-studied scheduling problem, which is in turn a restricted case of the well known problem STRIP PACKING. We present a collection of algorithms for RESTRICTED STRIP COVERING. We show that the problem is NP-hard and present an O(log log log n)-approximation algorithm. We also present better approximation or exact algorithms for some special cases, including when all intervals have equal width. For the general SEN-SOR COVER PROBLEM, we distinguish between cases in which elements have uniform or variable durations. The results depend on the structure of the region to be covered: We give a polynomial-time, exact algorithm for the uniform-duration case of RESTRICTED STRIP COVERING but prove that the uniformduration case for higher-dimensional regions is NP-hard. We give some more specific results for two-dimensional regions. Finally, we consider regions that are arbitrary sets, and we present an O(log n)-approximation algorithm for the most general case.
TimeSC, which is known as Restricted Strip Covering, was shown to be NP-hard by Buchsbaum et al. REF , who also gave an O(log log log n)-approximation algorithm.
Restricted strip covering and the sensor cover problem
SODA '07
Computer Science Mathematics