[论文解读] Approximation Algorithms for the Airport and Railway Problem

Cohen-Addad, Vincent, Le, Hung+2 more|arXiv (Cornell University)|Jan 1, 2023

Complexity and Algorithms in Graphs被引用 2

一句话总结

本文提出了一种随机化多项式时间算法，将边权平面图和无小图图嵌入到分枝宽为多对数的宿图中，其期望乘法畸变可任意接近 1。该嵌入使得在分枝宽较低的结构上使用动态规划，能够为诸如容量车辆路径规划和聚类等 NP 难问题设计准多项式时间近似方案。

ABSTRACT

In this paper, we present approximation algorithms for the airport and railway problem (AR) on several classes of graphs. The AR problem, introduced by [Anna Adamaszek et al., 2016], is a combination of the Capacitated Facility Location problem (CFL) and the network design problem. An AR instance consists of a set of points (cities) V in a metric d(.,.), each of which is associated with a non-negative cost f_v and a number k, which represent respectively the cost of establishing an airport (facility) in the corresponding point, and the universal airport capacity. A feasible solution is a network of airports and railways providing services to all cities without violating any capacity, where railways are edges connecting pairs of points, with their costs equivalent to the distance between the respective points. The objective is to find such a network with the least cost. In other words, find a forest, each component having at most k points and one open facility, minimizing the total cost of edges and airport opening costs. Adamaszek et al. [Anna Adamaszek et al., 2016] presented a PTAS for AR in the two-dimensional Euclidean metric ℝ² with a uniform opening cost. In subsequent work [Anna Adamaszek et al., 2018] presented a bicriteria 4/3 (2+1/α)-approximation algorithm for AR with non-uniform opening costs but violating the airport capacity by a factor of 1+α, i.e. (1+α)k capacity where 0 < α ≤ 1, a (2+k/(k-1)+ε)-approximation algorithm and a bicriteria Quasi-Polynomial Time Approximation Scheme (QPTAS) for the same problem in the Euclidean plane ℝ². In this work, we give a 2-approximation for AR with a uniform opening cost for general metrics and an O(log n)-approximation for non-uniform opening costs. We also give a QPTAS for AR with a uniform opening cost in graphs of bounded treewidth and a QPTAS for a slightly relaxed version in the non-uniform setting. The latter implies O(1)-approximation on graphs of bounded doubling dimensions, graphs of bounded highway dimensions and planar graphs in quasi-polynomial time.

研究动机与目标

• 设计一种高效嵌入方法，将平面图和无小图图嵌入到低分枝宽宿图中，实现接近最优的畸变。
• 通过允许期望畸变为 (1+ε)，突破常数畸变嵌入的 Ω(log n) 分枝宽下界。
• 统一并扩展无小图度量中网络设计、聚类和路由问题的近似方案。
• 利用低分枝宽宿图上的动态规划，实现对诸如容量车辆路径规划和设施选址等问题的 (1+ε)-近似。

提出的方法

• 构建从 n 个顶点的无小图图 G 到宿图 H 的随机嵌入 η: V(G) → V(H)，其中 H 的分枝宽在 1/ε 和 log n 中为多对数。
• 采用一种新颖的嵌入技术，确保 distH(η(u),η(v)) ≥ distG(u,v) 恒成立，且 E[distH(η(u),η(v))] ≤ (1+ε)distG(u,v)。
• 对边权进行归一化，以控制畸变并支持高效的动态规划。
• 在 H 的树分解上应用动态规划，利用已知的有界分枝宽图算法。
• 通过将嵌入边替换为 G 中的最短路径，将 H 上的解回传至 G。
• 采用一种框架，使得嵌入和回传步骤在 (1+ε) 因子内保持近似保证。

实验结果

研究问题

• RQ1 平面图和无小图图能否被嵌入到分枝宽为多对数的图中，实现 (1+ε) 期望畸变？
• RQ2 通过允许期望畸变，能否绕过常数畸变嵌入的 Ω(log n) 分枝宽下界？
• RQ3 此类嵌入能否为无小图度量中的 NP 难问题提供准多项式时间近似方案？
• RQ4 该框架如何应用于具有有界畸变的容量车辆路径规划和聚类问题？

主要发现

• 宿图 H 的分枝宽在 ε⁻¹、log n 和 G 的度量伸展值中为多项式，实现多对数分枝宽。
• 期望乘法畸变为 (1+ε)，且畸变不会低于原始值：distH(η(u),η(v)) ≥ distG(u,v) 恒成立。
• 该框架为容量车辆路径规划问题在准多项式时间内实现了 (1+ε)-近似。
• 对于容量 k-中位和设施选址问题，该算法在运行时间 2poly(1/ε, log n) 内实现了 (1+ε)-近似。
• 该嵌入是首个实现多对数分枝宽，同时逼近常数畸变嵌入理论下界 Ω(log n) 的方法。
• 该方法统一了无小图度量中多个 NP 难问题的近似方案，包括路由、聚类和网络设计。