AF: Small: Approximation Algorithms for Geometric Optimization

计算几何的方法将应用于设计、分析、实现和测试算法，以解决几个应用领域中出现的问题，包括几何网络优化、空中交通管理、传感器网络、机器人、几何建模和制造。主要的项目目标是发展几何问题的近似算法的基本进展。此外，该项目将努力促进和深化与应用领域和行业的研究人员和领域专家的合作，以便准确地制定他们的算法需求，并提供可用的算法工具、理论结果的见解和实验调查的软件。具体的问题包括：(a)几何网络优化：几何环境下的最优路由和网络设计，包括旅行推销员（TSP）变体、车辆路由、约束生成树、最小权重细分、各种约束下的最优路线规划和可生存网络设计；(b)空中交通管理：面对由天气和交通拥堵引起的动态和不确定约束、分区（负载平衡）和国家航空系统流量管理结构的优化，空域的最佳利用；(c)传感器网络和覆盖：传感器部署，定位，数据现场监测，以及固定或移动（机器人）传感器的覆盖。这些问题将在两个方面受到攻击：(1)使用形式化算法分析，试图证明最坏情况或平均情况下时间/空间的最紧可能边界（上和下），或问题的近似比率；(2)开发简单、快速、实用的解决技术，并进行实验比较。

The methodologies of computational geometry will be applied to design,analyze, implement, and test algorithms for problems that arise inseveral application areas, including geometric network optimization,air traffic management, sensor networks, robotics, geometric modeling,and manufacturing. The main project goal is the development offundamental advances in approximation algorithms for geometricproblems. Additionally, the project will strive to foster and deepencollaborations with researchers and domain experts in applicationareas and industry, in order to formulate their algorithmic needsprecisely and to make available algorithmic tools, insights fromtheoretical results, and software from experimental investigations.The specific repertoire of problems includes: (a) Geometric NetworkOptimization: optimal routing and network design in geometriccontexts, including traveling salesman (TSP) variants, vehiclerouting, constrained spanning trees, minimum-weight subdivisions,optimal route planning with various constraints, and survivablenetwork design; (b) Air Traffic Management: optimal use of airspacein the face of dynamic and uncertain constraints induced by weatherand traffic congestion, sectorization (load balancing), andoptimization of flow management structures for the National AirspaceSystem; (c) Sensor Networks and Coverage: sensor deployment,localization, data field monitoring, and coverage for stationary ormobile (robotic) sensors.The problems will be attacked on two fronts: (1) Use of formalalgorithmic analysis, attempting to prove the tightest possible bounds(upper and lower) on the worst-case or average-case time/space, orapproximation ratio for the problem; and (2) Development of solutiontechniques designed to be simple, fast, and practical, and which arecompared experimentally.