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心算法和网络流的最大流最小切定理。此外，这些主题出现在许多实际环境中，如容错通信网络的设计、道路交通的拥塞控制和社会网络的分析。在实际环境中出现的许多问题都是np困难的。这意味着不能在合理的运行时间内计算出最优解，对P .not取模。= NP猜想。因此，研究集中在近似算法上，即在最优解的保证因子范围内找到解的高效算法。在近似算法的设计和分析中，我使用的方法包括：组合优化（特别是匹配和网络流）的算法和理论，线性规划松弛的舍入，原始对偶方法，提升和项目方法以及动态规划。我计划在我最近的主要进展和期刊出版物的基础上，与我的研究生和合作者共同研究网络设计中一些突出的开放性问题。我的研究计划的三个关键模块总结如下。网络设计中的一个基本问题，称为NC-SNDP，是在给定的网络G中找到一个最小代价的子网络H，使H满足一些预先规定的节点连通性要求。我（和我的研究生）正在研究一个基本的开放问题：为NC-SNDP设计一个近似算法，该算法的保证与网络g的节点/边/终端的数量无关。(B)稀疏树和旅行推销员问题（TSP）生成树T的稀疏参数是T的边在切割中所占比例的所有切割的最大值。Goddyn推测，对于任何要求的瘦度，一个足够大的边连通性的图都有一个具有该瘦度的生成树。算法（多时）证明将给ATSP的近似算法带来重大进展。我的长期目标就是这样一个证明。我正在设计（和我的研究生）在特殊的图类中寻找瘦生成树的有效算法。该领域的一个关键开放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。我的直接目标是为这个问题的一个特殊情况推导出一个比2更好的近似保证，这个问题被称为森林增强问题（FAP），相对于通过Lift-and-Project方法获得的FAP的松弛。这将概括我和高（我的博士研究生）一起发表的结果。

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).