[论文解读] Sublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics

Yu Chen, Sanjeev Khanna+1 more|arXiv (Cornell University)|Mar 28, 2022

Complexity and Algorithms in Graphs被引用 1

一句话总结

本文提出了在一般度量空间中估计最小生成树（MST）和旅行商巡游（TSP）成本的亚线性空间与亚线性查询算法。它引入了“覆盖优势”这一概念，用于估计将MST转化为欧拉图的成本，从而在两轮图流中实现TSP的(1.96)-近似，在查询模型中使用Õ(n^1.5)次查询实现严格优于2的近似，即使在一般度量空间下也成立。

ABSTRACT

We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on n points. We start by exploring this estimation task in the regime of o(n) space, when the input is presented as a stream of all binom(n,2) entries of the metric in an arbitrary order (a metric stream). For any α ≥ 2, we show that both MST and TSP cost can be α-approximated using Õ(n/α) space, and moreover, Ω(n/α²) space is necessary for this task. We further show that even if the streaming algorithm is allowed p passes over a metric stream, it still requires Ω̃(√{n/α p²}) space. We next consider the well-studied semi-streaming regime. In this regime, it is straightforward to compute MST cost exactly even in the case where the input stream only contains the edges of a weighted graph that induce the underlying metric (a graph stream), and the main challenging problem is to estimate TSP cost to within a factor that is strictly better than 2. We show that in graph streams, for any ε > 0, any one-pass (2-ε)-approximation of TSP cost requires Ω(ε² n²) space. On the other hand, we show that there is an Õ(n) space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than 2 when the algorithm is given access to an n × n matrix that specifies pairwise distances between n points. The problem of MST cost estimation in this model is well-understood and a (1+ε)-approximation is achievable by Õ(n/ε^{O(1)}) queries. However, for estimating TSP cost, it is known that an analogous result requires Ω(n²) queries even for (1,2)-TSP, and for general metrics, no algorithm that achieves a better than 2-approximation with o(n²) queries is known. We make progress on this task by designing an algorithm that performs Õ(n^{1.5}) distance queries and achieves a strictly better than 2-approximation when either the metric is known to contain a spanning tree supported on weight-1 edges or the algorithm is given access to a minimum spanning tree of the graph. Prior to our work, such results were only known for the special cases of graphic TSP and (1,2)-TSP. In terms of techniques, our algorithms for metric TSP cost estimation in both streaming and query settings rely on estimating the cover advantage which intuitively measures the cost needed to turn an MST into an Eulerian graph. One of our main algorithmic contributions is to show that this quantity can be meaningfully estimated by a sublinear number of queries in the query model. On one hand, the fact that a metric stream reveals pairwise distances for all pairs of vertices provably helps algorithmically. On the other hand, it also seems to render useless techniques for proving space lower bounds via reductions from well-known hard communication problems. Our main technical contribution in lower bounds is to identify and characterize the communication complexity of new problems that can serve as canonical starting point for proving metric stream lower bounds.

研究动机与目标

• 设计用于在一般度量空间中估计MST和TSP成本的亚线性空间与查询复杂度算法。
• 在流媒体与查询模型中，建立近似MST和TSP成本的紧致空间与查询复杂度下界。
• 引入并利用新颖的“覆盖优势”概念作为亚线性设置下TSP成本估计的关键工具。
• 解决关于在o(n²)空间与查询下对一般度量空间中的TSP成本实现(2−ε)-近似这一开放问题。
• 为证明度量流下界，提供新的通信复杂度表征。

提出的方法

• 引入‘覆盖优势’作为使MST变为欧拉图的成本度量，捕捉MST与TSP成本之间的差距。
• 使用两轮流算法，重构边覆盖并利用MST结构，在1.96倍范围内近似TSP成本。
• 采用基于查询的算法，通过利用权重为1的边上的生成树或访问MST，执行Õ(n^1.5)次距离查询。
• 设计一种子程序框架，结合局部探索、BFS与子图重构，以亚线性查询估计覆盖优势。
• 通过针对度量流量身定制的新通信复杂度问题证明下界，避免从经典难题进行归约。
• 利用信息复杂度与概率论证，建立在图流中单轮(2−ε)-近似所需Ω(ε²n²)空间的下界。

实验结果

研究问题

• RQ1 在单轮度量流中，能否使用o(n²)空间实现对TSP成本的近似，其近似因子优于2？
• RQ2 是否存在一种亚线性查询算法，可在一般度量空间中实现TSP成本的(2−ε)-近似？
• RQ3 在单轮度量流中，MST与TSP估计的空间与近似比之间最优权衡为何？
• RQ4 覆盖优势概念能否用于在流媒体与查询模型中设计高效的TSP成本估计亚线性算法？
• RQ5 为证明度量流算法的强下界，需要哪些新的通信复杂度问题？

主要发现

• 一个˜O(n)空间的两轮算法在图流中实现了TSP成本的1.96-近似。
• 任何在图流中实现单轮(2−ε)-近似的TSP成本都需要Ω(ε²n²)空间，从而建立了紧致下界。
• 一种使用Õ(n^1.5)次查询的亚线性查询算法，在假设存在权重为1的边上的生成树或可访问MST的前提下，对一般度量空间中的TSP成本实现了严格优于2的近似。
• 本文建立了在单轮度量流中α-近似MST与TSP成本的近乎紧致空间下界˜Ω(n/α²)。
• 对于p轮流媒体算法，空间复杂度为˜Ω(√(n/αᵖ²))，表明多轮并不能显著降低空间需求。
• 本文识别并表征了新的通信问题，这些问题是证明度量流下界的标准起点。