Restricted Strip Covering and the Sensor Cover Problem

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.Comment: 14 pages, 6 figure

A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

Movers and Shakers: Kinetic Energy Harvesting for the Internet of Things

Numerous energy harvesting wireless devices that will serve as building blocks for the Internet of Things (IoT) are currently under development. However, there is still only limited understanding of the properties of various energy sources and their impact on energy harvesting adaptive algorithms. Hence, we focus on characterizing the kinetic (motion) energy that can be harvested by a wireless node with an IoT form factor and on developing energy allocation algorithms for such nodes. In this paper, we describe methods for estimating harvested energy from acceleration traces. To characterize the energy availability associated with specific human activities (e.g., relaxing, walking, cycling), we analyze a motion dataset with over 40 participants. Based on acceleration measurements that we collected for over 200 hours, we study energy generation processes associated with day-long human routines. We also briefly summarize our experiments with moving objects. We develop energy allocation algorithms that take into account practical IoT node design considerations, and evaluate the algorithms using the collected measurements. Our observations provide insights into the design of motion energy harvesters, IoT nodes, and energy harvesting adaptive algorithms.Comment: 15 pages, 11 figure

Online Resource Inference in Network Utility Maximization Problems

The amount of transmitted data in computer networks is expected to grow considerably in the future, putting more and more pressure on the network infrastructures. In order to guarantee a good service, it then becomes fundamental to use the network resources efficiently. Network Utility Maximization (NUM) provides a framework to optimize the rate allocation when network resources are limited. Unfortunately, in the scenario where the amount of available resources is not known a priori, classical NUM solving methods do not offer a viable solution. To overcome this limitation we design an overlay rate allocation scheme that attempts to infer the actual amount of available network resources while coordinating the users rate allocation. Due to the general and complex model assumed for the congestion measurements, a passive learning of the available resources would not lead to satisfying performance. The coordination scheme must then perform active learning in order to speed up the resources estimation and quickly increase the system performance. By adopting an optimal learning formulation we are able to balance the tradeoff between an accurate estimation, and an effective resources exploitation in order to maximize the long term quality of the service delivered to the users