[论文解读] The Message Complexity of Distributed Graph Optimization

Fabien Dufoulon, Shreyas Pai+3 more|arXiv (Cornell University)|Nov 24, 2023

Complexity and Algorithms in Graphs被引用 4

一句话总结

本文为四种基本问题——最大匹配（MaxM）、最小点覆盖（MVC）、最小支配集（MDS）和最大独立集（MaxIS）——的分布式图优化算法建立了紧致的消息复杂度界限。研究表明，在 CONGEST 模型中，精确计算 MVC、MDS 和 MaxIS 需要 Ω̃(n³) 条消息，而常数因子近似解则可通过 Θ̃(n²) 条消息实现。在随机图上，常数因子近似解可在多项式对数轮数内实现近线性 Õ(n) 的消息复杂度。

ABSTRACT

The message complexity of a distributed algorithm is the total number of messages sent by all nodes over the course of the algorithm. This paper studies the message complexity of distributed algorithms for fundamental graph optimization problems. We focus on four classical graph optimization problems: Maximum Matching (MaxM), Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). In the sequential setting, these problems are representative of a wide spectrum of hardness of approximation. While there has been some progress in understanding the round complexity of distributed algorithms (for both exact and approximate versions) for these problems, much less is known about their message complexity and its relation with the quality of approximation. We almost fully quantify the message complexity of distributed graph optimization by showing the following results: 1) Cubic regime: Our first main contribution is showing essentially cubic, i.e., Ω̃(n³) lower bounds (where n is the number of nodes in the graph) on the message complexity of distributed exact computation of Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). Our lower bounds apply to any distributed algorithm that runs in polynomial number of rounds (a mild and necessary restriction). Our result is significant since, to the best of our knowledge, this are the first ω(m) (where m is the number of edges in the graph) message lower bound known for distributed computation of such classical graph optimization problems. Our bounds are essentially tight, as all these problems can be solved trivially using O(n³) messages in polynomial rounds. All these bounds hold in the standard CONGEST model of distributed computation in which messages are of O(log n) size. 2) Quadratic regime: In contrast, we show that if we allow approximate computation then Θ̃(n²) messages are both necessary and sufficient. Specifically, we show that Ω̃(n²) messages are required for constant-factor approximation algorithms for all four problems. For MaxM and MVC, these bounds hold for any constant-factor approximation, whereas for MDS and MaxIS they hold for any approximation factor better than some specific constants. These lower bounds hold even in the LOCAL model (in which messages can be arbitrarily large) and they even apply to algorithms that take arbitrarily many rounds. We show that our lower bounds are essentially tight, by showing that if we allow approximation to within an arbitrarily small constant factor, then all these problems can be solved using Õ(n²) messages even in the CONGEST model. 3) Linear regime: We complement the above lower bounds by showing distributed algorithms with Õ(n) message complexity that run in polylogarithmic rounds and give constant-factor approximations for all four problems on random graphs. These results imply that almost linear (in n) message complexity is achievable on almost all (connected) graphs of every edge density.

研究动机与目标

• 本文旨在量化分布式算法在经典图优化问题上的消息复杂度。
• 旨在弥合对消息复杂度与近似质量之间关系的理解差距，特别是在分布式环境中的关系。
• 研究聚焦于四个 NP-难问题：MaxM、MVC、MDS 和 MaxIS，这些问题是近似难度谱系的代表。
• 探讨消息效率与近似质量之间的权衡，尤其在 CONGEST 和 LOCAL 模型中。
• 旨在建立不同场景下的紧致下界与上界：精确计算为立方级，近似为二次级，随机图为近线性。

提出的方法

• 作者通过信息论与通信复杂度技术，特别是 KT0 CONGEST 模型，证明了 ω(m) 的消息复杂度下界。
• 通过归约与下界构造，建立了 MVC、MDS 和 MaxIS 精确计算的 Ω̃(n³) 消息复杂度。
• 针对近似问题，提出一种随机化贪心算法，通过利用度数正则性与高概率浓度界限，在随机图上实现 Õ(n) 的消息复杂度。
• 证明了在 LOCAL 模型（即使消息大小无界）下，MaxM、MVC、MDS 和 MaxIS 的常数因子近似解仍需 Ω̃(n²) 消息复杂度，且该下界为无条件成立。
• 设计了一种基于随机边提议的消息高效算法，提议概率与 1/r（r = Δ/δ）成正比，确保匹配的稳定性并实现有界近似因子。
• 应用 McDiarmid 不等式，证明当 r = o(n^{1/4}/log n) 时，匹配大小具有高概率浓度，从而实现高置信度近似。

实验结果

研究问题

• RQ1 在 CONGEST 模型中，MVC、MDS 和 MaxIS 的精确分布式计算的消息复杂度是多少？
• RQ2 是否能以亚立方级消息复杂度实现 MaxM、MVC、MDS 和 MaxIS 的常数因子近似？
• RQ3 在消息大小无界的 LOCAL 模型中，近似解的消息复杂度是多少？
• RQ4 在随机图上，常数因子近似解是否可实现近线性消息复杂度？
• RQ5 近似因子能否实现高概率而非仅期望意义下的保证？在何种条件下可实现？

主要发现

• MVC、MDS 和 MaxIS 的精确计算消息复杂度为 Ω̃(n³)，该界基本紧致，因为 O(n³) 条消息已足够。
• 对于 MaxM、MVC、MDS 和 MaxIS 的常数因子近似解，即使在 LOCAL 模型中且允许任意多轮，仍需 Ω̃(n²) 条消息。
• 对于 MDS 和 MaxIS，任何优于特定常数的近似因子，其下界同样为 Ω̃(n²)。
• 一种随机化贪心算法在随机图上实现了 Õ(n) 的消息复杂度与 O(1) 轮数，适用于 p ≥ 16(log n)/n 的随机图。
• 该算法对最大匹配给出期望 O(r²) 因子的近似，其中 r = Δ/δ；当 r = O(1) 时（如正则图或随机图），该因子退化为 O(1)。
• 当 r = o(n^{1/4}/log n) 时，由于 McDiarmid 不等式带来的浓度界限，近似因子以高概率成立。