Approximation algorithms for NP-hard problems

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。许多重要的算法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和城市道路交通拥堵控制。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，研究主要集中在近似算法上，即找到在最优解的保证因子内的解的有效算法。 我当前和计划的研究重点是以下三个广泛的相互关联的主题。我在下面讨论其中两个主题，我的提案完整讨论了所有主题。 1. 近似最小成本网络的设计，包括旅行商问题（TSP）及其变体。 2. 符合节点连接要求的网络设计。 3. 非对称 TSP 的提升和投影方法及相关问题。 所有离散优化中最著名的问题是 TSP。最著名的算法结果是 1976 年 Christofides 提出的 3/2 近似算法。长期以来，人们猜测 TSP 存在 4/3 近似算法，并且存在称为 s-t 路径 TSP 的变体的 3/2 近似算法。 我正在研究的关于该主题的两个悬而未决的问题如下： (*) 改进了称为 GRAPHIC TSP 的重要特殊情况的 7/5 近似保证，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，加上 LP 解的支持图上改进的结构结果。 我研究的第二个主要主题涉及受节点连接要求影响的网络设计。网络设计的基本问题之一是找到给定网络G的最小成本子网H，使得H满足一些预先指定的连通性要求。 满足 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，并取得了许多里程碑式的成果。尽管已经进行了十多年的积极研究，但在涉及节点连接要求的类似问题上，进展却要慢得多。最近，在与 L.Vegh 共同撰写的一篇论文（Proc. IEEE FOCS 2013）中，我在该领域的一个基本问题上取得了重大进展：我们有一个用于最小成本 k 节点连通生成子图问题的 6 近似算法，假设节点数量至少为 k^4。我们的结果和技术为满足节点连接要求的网络设计开辟了许多新方向。我计划与研究生和博士后一起继续研究这些主题。 总之，我的研究议程的高级目标是在网络设计和组合优化相关领域取得重大进展。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可用的结果和技术。 像 TSP 这样的问题在所有现代社会中都普遍存在，包括加拿大；经济和基础设施以物流、运输、网络以及稀缺资源对关键任务的优化配置为基础。

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.