A $(1+1/\sqrt{2})$-近似算法用于多仓库旅行商问题

中文摘要 AI 辅助

度量旅行商问题（TSP）是组合优化中的一个基本问题，要求在一个度量图中找到一条覆盖所有客户的最小成本回路。度量多仓库旅行商问题（MD-TSP）是其自然扩展，其中图包含仓库和客户，目标是计算一组覆盖所有客户的最小成本回路集合，每条回路从同一仓库出发并返回该仓库。当仓库数量作为输入的一部分时，对Christofides--Serdyukov启发式算法的改编可得到近似比为$2$的算法。在本文中，我们提出了一种$(1+1/\sqrt{2})$-近似算法。与Christofides--Serdyukov启发式算法类似，我们的算法首先计算一个根生成森林（RSF），然后通过匹配来修正其奇度顶点，最后通过捷径化得到解。然而，我们并非使用最小成本RSF，而是通过针对自然割松弛的原对偶算法构造RSF。该算法以不同速率增长无根分量和包含所有仓库的分量，当对偶约束变紧时添加边。顶点标签记录客户首次连接到仓库的时间。双速增长提供了森林成本与两个标签相关项（这些项也出现在奇偶修正成本的界中）的联合界。通过将这两项的系数均设为$\sqrt{2}-1$来平衡，得到所声称的近似比。

英文摘要

The metric traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization that asks for a minimum-cost tour covering all clients in a metric graph. The metric multiple-depot TSP (MD-TSP) is a natural extension, where the graph contains depots and clients, and the objective is to compute a minimum-cost set of tours covering all clients, with each tour starting and ending at the same depot. When the number of depots is part of the input, an adaptation of the Christofides--Serdyukov heuristic yields an approximation ratio of $2$. In this paper, we introduce a $(1+1/\sqrt{2})$-approximation algorithm. Like the Christofides--Serdyukov heuristic, our algorithm first computes a rooted spanning forest (RSF), then a matching to correct its odd degrees, and finally obtains a solution by shortcutting. However, instead of using a minimum-cost RSF, we construct an RSF by a primal-dual algorithm for a natural cut relaxation. The algorithm grows rootless components and the component containing all depots at different rates, adding an edge when its dual constraint becomes tight. Vertex labels record the times at which clients first become connected to a depot. The two-speed growth provides a joint bound on the forest cost and two label-dependent terms that also arise in bounding the parity-correction cost. Balancing the coefficients of these two terms by setting both to $\sqrt{2}-1$ yields the claimed approximation ratio.

发表机构

• Kyung Hee University（庆熙大学）
• University of Electronic Science and Technology of China（电子科技大学）

机构由 AI 辅助整理，请以论文原文为准。