[论文解读] Local-search based approximation algorithms for mobile facility location problems

Sara Ahmadian, Zachary Friggstad+1 more|arXiv (Cornell University)|Jan 6, 2013

Facility Location and Emergency Management参考文献 29被引用 31

一句话总结

本文提出了首个基于局部搜索的移动设施定位（MFL）问题近似算法，对任意 ε > 0 实现 (3 + ε) 近似比。通过引入一种新型树状结构来引导局部搜索移动，并在每一步允许设施与目的地进行最优重匹配，该方法达到了与 k-中位问题最佳已知近似保证相同的性能，并通过新方法证明了其近似比在 o(1) 因子范围内的紧致性。

ABSTRACT

We consider the mobile facility location (MFL) problem. We are given a set of facilities and clients located in a common metric space G = (V, c). The goal is to move each facility from its initial location to a destination (in V) and assign each client to the destination of some facility so as to minimize the sum of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs.We give the first local-search based approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is (3 + e)-approximation for this problem for any constant e > 0 using local search. The previous best guarantee for MFL was an 8-approximation algorithm due to Friggstad and Salavatipour [12] based on LP-rounding. Our guarantee matches the best-known approximation guarantee for the k-median problem. Since there is an approximation-preserving reduction from the k-median problem to MFL, any improvement of our result would imply an analogous improvement for the k-median problem. Furthermore, our analysis is tight (up to o(1) factors) since the tight example for the local-search based 3-approximation algorithm for k-median can be easily adapted to show that our local-search algorithm has a tight approximation ratio of 3. Our results extend to the weighted generalization wherein each facility i has a non-negative weight wi and the movement cost for i is wi times the distance traveled by i.In contrast to the k-median problem, the local search procedure that moves, at each step, a constant number of facilities (to chosen destinations) and assigns each client to the nearest destination, is known to have an unbounded locality gap. Our local-search algorithm is a natural and simple variant, where we only select the destinations of the facilities in each step and optimally rematch the facilities to these destinations (which might entail moving all facilities). One of the chief novelties in the analysis is that in order to generate a suitable collection of local-search moves whose resulting inequalities yield the desired bound on the cost of a local-optimum, we define a tree-like structure that (loosely speaking) functions as a recursion tree, using which we spawn off local-search moves by exploring this tree to a constant depth.

研究动机与目标

• 开发一种基于局部搜索的移动设施定位（MFL）问题近似算法，允许设施重新定位以降低客户分配成本和移动成本。
• 弥合 MFL 问题与 k-中位问题之间最佳已知近似保证的差距，后者已知可归约为 MFL 问题。
• 通过引入一种改进的局部搜索过程并结合最优重匹配，克服标准局部搜索（仅移动常数个设施）存在的无界局部性间隙。
• 将该算法扩展至加权 MFL 变种，其中设施移动成本与设施权重成比例。
• 通过构建一种基于树状递归结构的合适局部搜索移动集合，提供近似比的紧致分析。

提出的方法

• 该算法采用局部搜索框架，每一步选择一组设施目的地，并将所有设施最优地重新匹配到这些目的地，而非仅移动少数设施。
• 构建一种新型树状结构以引导局部搜索移动的生成，模拟一个探索至常数深度的递归树，以寻找潜在改进。
• 该树状结构使得能够生成一组局部搜索移动，其成本不等式被用于界定局部最优解的成本。
• 分析利用了 k-中位问题局部搜索算法中的任何改进均可被适配至 MFL 的事实，从而确保 (3 + ε) 近似比的紧致性。
• 通过按设施权重缩放移动成本，该方法处理了加权 MFL 变种，同时保持相同的近似保证。
• 该算法通过不限制每次迭代中移动的设施数量，而是对选定目的地重新最优分配所有设施，从而避免了标准局部搜索的无界局部性间隙。

实验结果

研究问题

• RQ1 基于局部搜索的算法能否实现移动设施定位问题的最佳已知近似比？
• RQ2 为何标准局部搜索（仅移动常数个设施）在 MFL 问题中失效，以及如何解决这一问题？
• RQ3 能否通过局部搜索方法在 MFL 问题中实现或超越 k-中位问题的近似保证？
• RQ4 解空间的哪些结构特性使得 MFL 问题中局部搜索算法的分析能够达到紧致性？
• RQ5 如何利用树状结构生成有效的局部搜索移动，以获得有界近似比？

主要发现

• 所提出的局部搜索算法对任意 ε > 0 实现了移动设施定位问题的 (3 + ε) 近似比，与 k-中位问题的最佳已知近似保证一致。
• 该算法的近似比在 o(1) 因子范围内是紧致的，因为 k-中位问题局部搜索 3-近似紧致示例可被适配至 MFL 问题。
• 该方法通过在每一步对选定目的地最优重匹配所有设施，克服了标准局部搜索的无界局部性间隙。
• 分析中使用的树状结构使得能够系统性地生成局部搜索移动，从而产生用于成本界定的不等式。
• 该算法可自然扩展至加权 MFL 变种，在移动成本按设施权重缩放时，仍保持 (3 + ε) 近似保证。
• 任何对 (3 + ε) 近似比的改进都将意味着 k-中位问题的相应改进，因为存在保持近似比的归约关系。