[论文解读] Online network design algorithms via hierarchical decompositions

Seeun William Umboh|arXiv (Cornell University)|Jan 4, 2015

Optimization and Search Problems参考文献 19被引用 19

一句话总结

本文提出了一种新颖的确定性在线网络设计框架，通过将网络分解为分层良好分离的树（HSTs），实现了对关键问题（如斯坦纳网络、租购问题、连通设施位置和奖赏收集斯坦纳森林）的更简单分析，并在竞争比方面达到或优于以往结果，同时统一了各类问题的分析方法。

ABSTRACT

We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design.In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings --- embed the entire graph into a tree at the beginning and then solve the problem on the tree --- and results in randomized O(log n)-competitive algorithms, where n is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get O(log k) competitive ratios, where k is the number of terminals.In this paper, we apply our approach to obtain deterministic O(log k)-competitive online algorithms for the following problems.1. Steiner network with edge duplication. Previously, only a randomized O(log n)-competitive algorithm was known.2. Rent-or-buy. Previously, only deterministic O(log2k)-competitive and randomized O(log k)-competitive algorithms by Awerbuch, Azar and Bartal (Theoretical Computer Science 2004) were known.3. Connected facility location. Previously, only a randomized O(log2k)-competitive algorithm of San Felice, Williamson and Lee (LATIN 2014) was known.4. Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (ICALP 2011) and give a simpler analysis.Our competitive ratios are optimal up to constant factors as these problems capture the online Steiner tree problem which has a lower bound of Ω(log k).

研究动机与目标

• 开发一种统一的、确定性的在线网络设计方法，以获得更优的竞争比。
• 通过用终端的HST嵌入替代对偶拟合，简化在线网络算法的分析。
• 为斯坦纳网络、租购问题和连通设施位置等基本问题实现最优竞争比。
• 使用单一、可推广的技术，匹配或优于现有的随机化或复杂确定性算法。

提出的方法

• 将终端集合嵌入到分层良好分离的树（HST）中，以实现算法解与最优解之间的成本比较。
• 通过将在线算法的成本归因于HST嵌入上最优解的成本，利用树的分层结构进行成本分析。
• 采用确定性的贪心类算法进行在线决策，避免了复杂对偶解构造的需要。
• 将基于HST的分析技术应用于多种网络设计问题，统一了其竞争比分析。
• 利用已知的HST嵌入性质，控制原始图与树之间距离的失真，确保O(log k)的竞争比。
• 通过将问题与在线斯坦纳树问题关联，证明竞争比在常数因子内是最优的。

实验结果

研究问题

• RQ1 能否开发一种确定性的在线网络设计框架，避免对偶拟合的复杂性，同时实现与随机化算法相当或更优的竞争比？
• RQ2 HST嵌入在多大程度上可用于统一分析多种在线网络设计问题？
• RQ3 能否通过更简单、确定性的方法，为连通设施位置和租购问题等实现O(log k)的竞争比？
• RQ4 鉴于这些问题是在线斯坦纳树问题的推广，O(log k)的竞争比是否对这些问题是紧的？

主要发现

• 本文为允许边重复的斯坦纳网络问题实现了确定性的O(log k)-竞争比算法，与最佳已知随机化结果竞争比相同，但分析更简单。
• 对于租购问题，该方法给出了确定性的O(log k)-竞争比算法，优于先前的确定性O(log²k)结果。
• 连通设施位置问题通过确定性的O(log k)-竞争比算法解决，竞争比与最佳已知结果一致，但证明显著简化。
• 奖赏收集斯坦纳森林问题通过O(log k)-竞争比的确定性算法解决，与先前最佳结果一致，但分析更简洁。
• 该框架证明，对于所有这些问题，O(log k)是竞争比的最优值（在常数因子内），因为它们是已知具有Ω(log k)下界在线斯坦纳树问题的推广。