限制带覆盖与传感器覆盖问题
Restricted Strip Covering and the Sensor Cover Problem
摘要：
给定一组具有覆盖基本区域的持续时间的对象（作业），我们能否调度作业以最大化原始区域保持覆盖的持续时间？我们把这个问题称为传感器盖问题。在用传感器覆盖区域的背景下出现该问题。例如，假设您希望通过放置在各个固定位置的传感器来监视围栏上的活动。每个传感器都有一个范围和有限的电池寿命。问题是安排何时打开传感器，以便尽可能长时间地全面监控围栏。这个一维问题涉及到实数直线上的区间。将一个持续时间关联到每一个会产生一组空间和时间上的矩形，每个矩形由一对固定的水平端点和高度指定。目标是为每个矩形分配一个位置，以最大化跨越间隔完全覆盖的高度。我们称之为一维问题限制带覆盖。如果我们将覆盖约束替换为装箱约束，则问题等同于动态存储分配，这是一个调度问题，它是带装箱问题的限制情况。证明了带约束覆盖问题是NP难的，并给出了一个O（log log n）-近似算法.我们提出了更好的近似或精确算法的一些特殊情况。对于均匀持续时间的情况下，限制带覆盖，我们给出了一个多项式时间，精确的算法，但证明了均匀持续时间的情况下，高维区域是NP-困难的。最后，我们考虑区域是任意集，我们提出了一个O（log n）-近似算法。
英文摘要：
Given a set of objects with durations (jobs) that cover a base region, can we schedule the jobs to maximize the duration the original region remains covered? We call this problem the sensor cover problem. This problem arises in the context of covering a region with sensors. For example, suppose you wish to monitor activity along a fence by sensors placed at various fixed locations. Each sensor has a range and limited battery life. The problem is 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 position to each rectangle 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, the problem is identical to dynamic storage allocation, a scheduling problem that is a restricted case of the strip packing problem. We show that the restricted strip covering problem is NP-hard and present an O(log log n)-approximation algorithm. We present better approximations or exact algorithms for some special cases. For the uniform-duration case of restricted strip covering we give a polynomial-time, exact algorithm but prove that the uniform-duration case for higher-dimensional regions is NP-hard. Finally, we consider regions that are arbitrary sets, and we present an O(log n)-approximation algorithm.
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