Approximation algorithms for NP-hard problems

NP 困难问题的近似算法

• 批准号: RGPIN-2014-04351
• 负责人: Cheriyan, Joseph
• 金额: $ 3.35万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=634862
• 关键词: Approximation; algorithms; NP; hard; problems

Network design, network flows, and graph connectivity occur as core topics in Theoretical Computer Science, Operations Research, and Combinatorial Optimization. Many important algorithmic 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 and congestion control for urban road traffic. 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.My current and planned research focuses on the following three broad interlocking themes. I discuss two of these topics below, and my proposal discusses all the topics in full.1. Design of approximately minimum-cost networks, including the Traveling Salesman Problem (TSP) and its variants.2. Design of networks subject to node-connectivity requirements.3. Lift-and-Project methods for the Asymmetric TSP and related problems.The most famous problem in all of discrete optimization is the TSP. The best known algorithmic result is the 3/2-approximation algorithm due to Christofides from 1976. It has long been conjectured that there exists a 4/3-approximation algorithm for the TSP, and that there exists a 3/2-approximation algorithm for a variant called the s-t path TSP.Two of the outstanding open questions on this topic that I am researching are the following:(*) Improve on the approximation guarantee of 7/5 for an important special case called the GRAPHIC TSP, possibly based on a combination of LP-rounding techniques and ear-decomposition techniques.(*) Improve on the approximation guarantee of 8/5 for the s-t path TSP, possibly based on LP-rounding techniques, coupled with improved structural results on the support graph of LP solutions.The second broad theme of my research addresses the design of networks subject to node-connectivity requirements. One of the basic problems in network design is to find a minimum-cost sub-network H of a given network G such that H satisfies some pre-specified connectivity requirements. The area of minimum-cost network design subject to EDGE-connectivity requirements flourished in the 1990s, and there were a number of landmark results. Progress has been much slower on similar problems with NODE-connectivity requirements, despite more than a decade of active research. Very recently, in a paper co-authored with L.Vegh (Proc. IEEE FOCS 2013), I have obtained a major advance on a fundamental problem in this area: we have a 6-approximation algorithm for the minimum-cost k-node connected spanning subgraph problem, assuming that the number of nodes is at least k^4. Our results and techniques have opened up many new directions in the design of networks subject to node-connectivity requirements. I plan to continue research on these topics, together with graduate students and postdocs.In summary, the high-level goal of my research agenda is to provide significant advances in the areas of Network Design and related areas of Combinatorial Optimization. This has the potential to improve the results and techniques available to all researchers who work in this core area of the computational sciences. Problems such as the TSP are ubiquitous in all modern societies, including Canada; the economy and infrastructure are based on logistics, transport, networks, and on the optimal allocation of scarce resources to critical tasks.

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。许多重要的算法范例都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和城市道路交通的拥塞控制。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，有效的算法，找到解决方案，是在一个保证因素的最佳解决方案。我目前和计划的研究集中在以下三个广泛的联锁主题。下面我将讨论其中的两个主题，我的建议将全面讨论所有主题。近似最小成本网络的设计，包括旅行商问题（TSP）及其变种.设计符合节点连接要求的网络。求解非对称TSP及相关问题的提升投影方法。在所有离散优化中最著名的问题是TSP。最著名的算法结果是3/2近似算法由于Christofides从1976年。长期以来，人们一直认为存在一个4/3近似算法的TSP，并存在一个3/2近似算法的变种称为s-t路径TSP。两个突出的开放问题，我正在研究这个主题是：（*）改进的近似保证7/5的一个重要的特殊情况称为图形TSP，可能基于LP舍入技术和耳分解技术的组合。（*）改进s-t路径TSP的8/5近似保证，可能基于LP舍入技术，再加上LP解的支持图上的改进结构结果。我研究的第二个广泛主题涉及节点连通性要求的网络设计。网络设计中的一个基本问题是找到给定网络G的最小代价子网络H，使得H满足某些预先指定的连通性要求。符合EDGE连接要求的最低成本网络设计领域在20世纪90年代蓬勃发展，并取得了一些具有里程碑意义的成果。尽管进行了十多年的积极研究，但在节点连接要求的类似问题上，进展要慢得多。最近，在与L.Vegh合著的一篇论文（Proc. IEEE FOCS 2013）中，我在这个领域的一个基本问题上取得了重大进展：我们有一个6-近似算法来解决最小成本k-节点连接生成子图问题，假设节点数至少为k^4。我们的研究结果和技术开辟了许多新的方向，在网络设计的节点连接的要求。我计划与研究生和博士后一起继续研究这些主题。总之，我的研究议程的高层次目标是在网络设计和组合优化的相关领域提供重大进展。这有可能改善所有在计算科学这一核心领域工作的研究人员的结果和技术。像TSP这样的问题在包括加拿大在内的所有现代社会都是普遍存在的;经济和基础设施都建立在物流、运输、网络以及将稀缺资源最佳分配给关键任务的基础上。