Approximation Algorithms for NP-Hard Problems

网络设计、网络流和图连通性是理论计算机科学、运筹学和组合优化的核心主题。重要的算法和结构范例是在这些主题的背景下发展起来的，例如最小生成树的贪婪算法和网络流的最大流最小割定理。此外，这些主题出现在许多实际情况下，如容错通信网络的设计，道路交通的拥塞控制，以及社交网络的分析。在实际应用中，许多问题都是NP难的。这意味着最优解决方案不能在合理的运行时间内计算，以P. NP猜想因此，研究集中在近似算法上，即，有效算法，找到在最优解的保证因子内的解。 在近似算法的设计和分析中，我使用的方法包括：组合优化的算法和理论（特别是匹配和网络流），线性规划松弛的舍入，原始对偶方法，提升和投影方法和动态规划。 我计划与我的研究生和合著者一起，在我最近的主要进展和期刊出版物的基础上，解决网络设计中一些悬而未决的问题。 我的研究计划的三个关键模块总结如下。 (A)满足节点连通性要求的网络设计 网络设计中的一个基本问题，称为NC-SNDP，是找到给定网络G的最小成本子网络H，使得H满足某些预先指定的节点连接要求。我（和我格拉德生）正在解决一个基本的开放问题：设计一个NC-SNDP的近似算法，其保证与网络G的节点/边/终端的数量无关。 (B)稀疏树与旅行商问题（TSP） 生成树T的稀疏度参数是T的边在割中的比例在所有割上的最大值。 Goddyn指出，对于任何所需的稀疏度，一个足够大的边连通度的图都有一个具有该稀疏度的生成树。一个算法（多时间）的证明将使ATSP的近似算法取得重大进展。我的长期目标就是这样的证明。我正在设计（与我格拉德生）有效的算法，用于在特殊类的图中找到瘦生成树。 (C)组合优化的提升-投影系统 该领域的一个关键问题是改进最小代价2-边连通生成子图（2-ECSS）问题的2的近似保证。我的直接目标是得到一个近似保证优于两个这个问题的一个特殊情况下，称为森林扩增问题（FAP）相对于通过提升和项目方法获得的FAP放松。这将推广我和高（我完成的博士学位）一起发表的结果。学生）。

Network design, network flows, and graph connectivity are core topics in Theoretical Computer Science, Operations Research, and Combinatorial Optimization. Important algorithmic and structural paradigms were developed in the context of these topics, such as the greedy algorithm for minimum spanning trees and the max-flow min-cut theorem for network flows. Moreover, these topics arise in many practical contexts such as the design of fault-tolerant communication networks, congestion control for road traffic, and the analysis of social networks. Many of the problems arising in practical contexts are NP-hard. This means that optimal solutions cannot be computed in a reasonable running time, modulo the P .not.= NP conjecture. Hence, research has focused on approximation algorithms, i.e., efficient algorithms that find solutions that are within a guaranteed factor of the optimal solution. In the design and analysis of approximation algorithms, I am using methods such as: algorithms and theory from combinatorial optimization (in particular, matchings and network flows), rounding of linear-programming relaxations, the primal-dual method, lift-and-project methods, and dynamic programming. I plan to attack some outstanding open questions in network design jointly with my graduate students and co-authors, building on my recent major advances and journal publications. Three key modules of my research program are summarized below. (A) Network Design for Node-Connectivity Requirements A basic problem in network design, called the NC-SNDP, is to find a minimum-cost sub-network H of a given network G such that H satisfies some prespecified NODE-connectivity requirements. I am attacking (with my grad students) a fundamental open problem: design an approximation algorithm for NC-SNDP whose guarantee is independent of the number of nodes/edges/terminals of the network G. (B) Thin trees and the Traveling Salesman Problem (TSP) The thinness parameter of a spanning tree T is the maximum over all cuts of the proportion of the edges of T in the cut. Goddyn conjectured that for any required thinness, a graph of sufficiently large edge-connectivity has a spanning tree with that thinness. An algorithmic (poly-time) proof would give major advances on approximation algorithms for ATSP. My long-term goal is such a proof. I am designing (with my grad students) efficient algorithms for finding thin spanning trees in special classes of graphs. (C) Lift-and-Project Systems for Combinatorial Optimization A key open question in the area is to improve on the approximation guarantee of two for the minimum-cost 2-edge connected spanning subgraph (2-ECSS) problem. My immediate goal is to derive an approximation guarantee better than two for a special case of this problem called the Forest Augmentation Problem (FAP) relative to a relaxation of FAP obtained via Lift-and-Project methods. This will generalize results that I have published with Gao (my completed Ph.D. student).