[论文解读] Low congestion cycle covers and their applications

Merav Parter, Eylon Yogev|arXiv (Cornell University)|Jan 6, 2019

Complexity and Algorithms in Graphs被引用 5

一句话总结

本文引入了低拥塞环覆盖——在无桥图中，每个环长度较短（O(D)），且每条边仅出现在O(1)个环中的环集合——证明了其存在性，并提供了高效的构造方法。关键贡献是构造了一个(d, c)-环覆盖，其中d = O(D)，c = O(1)，从而实现了在拜占庭故障下的高效容错分布式计算。

ABSTRACT

A cycle cover of a bridgeless graph G is a collection of simple cycles in G such that each edge e appears on at least one cycle. The common objective in cycle cover computation is to minimize the total lengths of all cycles. Motivated by applications to distributed computation, we introduce the notion of low-congestion cycle covers, in which all cycles in the cycle collection are both short and nearly edge-disjoint. Formally, a (d, c)-cycle cover of a graph G is a collection of cycles in G in which each cycle is of length at most d and each edge participates in at least one cycle and at most c cycles.A-priori, it is not clear that cycle covers that enjoy both a small overlap and a short cycle length even exist, nor if it is possible to efficiently find them. Perhaps quite surprisingly, we prove the following: Every bridgeless graph of diameter D admits a (d, c)-cycle cover where d = O(D) and c = O(1). That is, the edges of G can be covered by cycles such that each cycle is of length at most O(D) and each edge participates in at most O(1) cycles. These parameters are existentially tight up to polylogarithmic terms.Furthermore, we show how to extend our result to achieve universally optimal cycle covers. Let Ce is the shortest cycle that covers e, and let OPT(G) = maxeϵG |Ce|. We show that every bridgeless graph admits a (d, c)-cycle cover where d = O(OPT(G)) and c = O(1).We demonstrate the usefulness of low congestion cycle covers in different settings of resilient computation. For instance, we consider a Byzantine fault model where in each round, the adversary chooses a single message and corrupt in an arbitrarily manner. We provide a compiler that turns any r-round distributed algorithm for a graph G with diameter D, into an equivalent fault tolerant algorithm with r·poly(D) rounds.

研究动机与目标

• 为解决在无桥图中构造环覆盖的挑战，平衡环的短长度与边的低重叠，以应用于弹性分布式系统。
• 证明存在满足d = O(D)且c = O(1)的(d, c)-环覆盖，其中D为图的直径，确保环短且拥塞有界。
• 通过以每条边的最短环覆盖长度为最优性基准，将结果扩展为普遍最优环覆盖，实现d = O(OPT(G))且c = O(1)，其中OPT(G)为覆盖每条边的最短环长度。
• 展示低拥塞环覆盖在基于编译器的拜占庭分布式算法容错中的实际效用。

提出的方法

• 将(d, c)-环覆盖定义为一组环，其中每个环的长度至多为d，且每条边至多出现在c个环中。
• 利用图的结构性质和环分解技术，证明任意直径为D的无桥图均存在满足d = O(D)且c = O(1)的(d, c)-环覆盖。
• 通过以覆盖每条边的最短环作为最优性基准，将构造方法扩展至实现d = O(OPT(G))且c = O(1)。
• 设计一个编译器，将图G上的任意r轮分布式算法转换为在拜占庭敌手模型下具有r·poly(D)轮的容错版本。
• 利用低拥塞环覆盖作为骨干，协调可靠的消息传播，即使每轮有单条消息被敌手篡改也能保持正确性。

实验结果

研究问题

• RQ1 在所有无桥图中，能否存在同时短且近乎边不相交的低拥塞环覆盖？
• RQ2 此类环覆盖中，环长度与边拥塞之间的最优权衡是什么？
• RQ3 能否在小直径图中高效构造此类环覆盖？
• RQ4 如何利用低拥塞环覆盖构建在拜占庭故障下的容错分布式算法？
• RQ5 是否可能实现普遍最优环覆盖，使得环长度在最短覆盖环长度的常数倍范围内？

主要发现

• 任意直径为D的无桥图均存在满足d = O(D)且c = O(1)的(d, c)-环覆盖，证明了短且低重叠环覆盖的存在性。
• 环覆盖参数在多对数因子范围内为存在性紧致，意味着无法实现更优的渐近权衡。
• 存在普遍最优环覆盖，满足d = O(OPT(G))且c = O(1)，其中OPT(G)为覆盖任意边的最短环的最大长度。
• 该构造支持一个编译器，可将任意r轮分布式算法转换为具有r·poly(D)轮的拜占庭容错版本。
• 环覆盖结构确保即使敌手每轮篡改一条消息，算法仍能通过冗余和环覆盖的协调保持正确性。