Approximation Algorithms for NP-Hard Problems

NP 困难问题的近似算法

• 批准号: RGPIN-2019-04197
• 负责人: Cheriyan, Joseph
• 金额: $ 3.5万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=689640
• 关键词: Approximation; Algorithms; NP; Hard; Problems

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).**

网络设计、网络流和图连接是理论计算机科学、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开发的，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环境中，例如容错通信网络的设计、道路交通拥塞控制以及社交网络分析。实际情况中出现的许多问题都是 NP 困难的。这意味着无法在合理的运行时间内计算出最优解，以 P .not.= NP 猜想为模。因此，研究的重点是近似算法，即找到在最优解的保证因子内的解决方案的有效算法。******在近似算法的设计和分析中，我使用的方法包括：算法和组合优化理论（特别是匹配和网络流）、线性规划松弛的舍入、原对偶方法、提升和投影方法以及动态规划。******我计划攻击一些与我一起解决网络设计中的突出开放问题研究生和合著者，以我最近的主要进展和期刊出版物为基础。******我的研究计划的三个关键模块总结如下。******(A) 节点连接要求的网络设计****** 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 H，使得 H 满足一些预先指定的节点连接要求。我正在（和我的研究生）解决一个基本的开放问题：为 NC-SNDP 设计一个近似算法，其保证独立于网络的节点/边/终端的数量 G.******(B) Thin树和旅行商问题（TSP）*****生成树 T 的稀疏参数是所有割中 T 的边在割中所占比例的最大值。 Goddyn 推测，对于任何所需的稀疏性，足够大的边连通性图都具有具有该稀疏性的生成树。算法（多时间）证明将为 ​​ATSP 的近似算法带来重大进步。我的长期目标就是这样的证明。我正在（与我的研究生）设计有效的算法，用于在特殊类别的图中查找稀疏生成树。******(C) 用于组合优化的提升和投影系统******一个关键的开放问题该领域的目标是改进最小成本2边连通生成子图（2-ECSS）问题的两个近似保证。我的近期目标是针对称为森林增强问题 (FAP) 的问题的特殊情况，相对于通过提升和投影方法获得的 FAP 松弛，得出优于 2 的近似保证。这将概括我与高（我的已完成博士生）发表的结果。**