1 research outputs found
Low Congestion Cycle Covers and their Applications
A cycle cover of a bridgeless graph is a collection of simple cycles in
such that each edge 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
-cycle cover of a graph is a collection of cycles in in which
each cycle is of length at most and each edge participates in at least one
cycle and at most 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 admits a -cycle cover
where and . These parameters are
existentially tight up to polylogarithmic terms. Furthermore, we show how to
extend our result to achieve universally optimal cycle covers. Let is the
length of the shortest cycle that covers , and let . We show that every bridgeless graph admits a -cycle cover where and . 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 -round distributed algorithm for a graph
with diameter , into an equivalent fault tolerant algorithm with rounds.Comment: arXiv admin note: substantial text overlap with arXiv:1712.0113