低拥塞圈覆盖及其应用
Low Congestion Cycle Covers and their Applications
摘要：
无桥图$G$的圈覆盖是$G$中的简单圈的集合，使得每个边$G$至少出现在一个圈上。圈覆盖计算的共同目标是最小化所有圈的总长度。受分布式计算应用的启发，我们引入了\emph{low-congestion}循环覆盖的概念，其中循环集合中的所有循环都是\emph{short}和近似\emph{edge-disjoint}。形式上，图$G$的$(d,c)$ -圈覆盖是$G$中的圈的集合，其中每个圈的长度至多为$d$，并且每条边参与至少一个圈和至多$c$圈。先验地，不清楚的是，甚至存在享有小重叠和短循环长度的循环覆盖，也不清楚是否可以有效地找到它们。也许相当令人惊讶的是，我们证明了以下几点：每一个无桥图直径$D$承认$(d,c)$ -循环覆盖，其中$d = \tilde{O}(D)$和$c=\tilde{O}(1)$。这些参数在多对数项下是存在紧的。此外，我们展示了如何扩展我们的结果，以实现普遍最优的循环覆盖。令$C_e$是覆盖$e$的最短循环的长度，令$OPT(G)= \max_{e \in G} C_e$。证明了每个无桥图都有$(d,c)$ -圈覆盖，其中$d = \tilde{O}(OPT(G))$和$c=\tilde{O}(1)$ .我们证明了低拥塞周期覆盖在不同的弹性计算设置的有用性。例如，我们考虑一个拜占庭故障模型，在每一轮中，对手选择一个单一的消息，并以任意的方式损坏。我们提供了一个编译器，把任何$r$轮分布式算法的图$G$直径$D$，到一个等效的容错算法与$r\cdot poly(D)$轮。
英文摘要：
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 \emph{low-congestion} cycle covers, in which all cycles in the cycle collection are both \emph{short} and nearly \emph{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 $d = \tilde{O}(D)$ and $c=\tilde{O}(1)$. These parameters are existentially tight up to polylogarithmic terms. Furthermore, we show how to extend our result to achieve universally optimal cycle covers. Let $C_e$ is the length of the shortest cycle that covers $e$, and let $OPT(G)= \max_{e \in G} C_e$. We show that every bridgeless graph admits a $(d,c)$-cycle cover where $d = \tilde{O}(OPT(G))$ and $c=\tilde{O}(1)$. 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 $r\cdot poly(D)$ rounds.
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