New Directions in Approximation Algorithms

该提案寻求继续资助PI在他的主要研究领域近似算法的工作。以前由PI的研究小组资助的nsf工作已经成功地制定并解决了几个有趣的近似问题，包括批量购买网络设计，$k$最小生成树，群斯坦纳树和度有界最小生成树问题。我们在这一领域的贡献包括困难问题的数学规划松弛和使用确定性和随机舍入技术对这些松弛进行舍入的新方法，以及新的问题公式。本提案的重点是通过更接近现实世界的约束，在近似算法的脉络中解决计算困难的问题。具体来说，我们建议将整合、异质性、竞争、不确定性和混合输入模型纳入经典问题。提案报告了这些方向的初步成果和正在进行的调查，并拟订了具体的新问题和办法。该提案的智力价值是在扩大其范围和适用性方面推动了近似算法的前沿，以及发现新底层技术的潜力。该提案还补充了一项教育和推广计划，使PI继续参与更广泛的教育目标，如参与调查、辅导和教学讲习班，以及新课程和讲义的编写。该提案的更广泛影响包括研究生在该研究领域的培训和安置，以及旨在向更广泛的受众（包括四年制大学讲师）传播PI研究的广泛教育计划。

This proposal seeks continued funding for the PI's work in his principal research area of Approximation Algorithms. Previous NSF-funded work by the PI's research group has successfully formulated and solved several interesting problems for approximation including the Buy-at-Bulk Network Design, $k$-Minimum Spanning Tree, Group Steiner Tree, and the Degree-bounded Minimum Spanning Tree problems. Our contributions to this area include mathematical programming relaxations of hard problems and novel ways to round such relaxations using deterministic and randomized rounding techniques, as well as new problem formulations.The focus of this proposal is to address the solution of computationally hard problems in the vein of approximation algorithms by moving closer to more real-world constraints.Specifically, we propose work in directions involving incorporating integration, heterogeneity, competition, uncertainty, and hybrid input models into classical problems. The proposal reports on preliminary successes and ongoing investigation in each of these directions, and formulates specific new problems and approaches. The intellectual merit of the proposal is to push the frontiers of approximation algorithms in terms of expanding its scope and applicability, as well as the potential for discovery of new underlying techniques.The proposal is complemented with an educational and outreach plan that continues the PI's involvement in broader educational goals such as participation in surveys, tutorials and teaching workshops, as well as new course and lecture note development. Thebroader impact of the proposal include graduate student training and placement in this research area as well as an extensive education plan aimed at increased dissemination of the PI's research to a broader audience including four-year college lecturers.