图的相识时间
Acquaintance Time of a Graph
摘要：
我们定义了连通图的以下参数。对于给定的图$G$，我们在$G$的每个顶点放置一个代理。每一对共享共同边缘的代理被宣布为相互认识。在每一轮中，我们选择$G$的一些匹配（不一定是最大匹配），并且对于匹配中的每个边缘，该边缘上的代理交换位置。在交换之后，再次，共享共同边缘的每对代理变得熟悉，并且该过程继续。我们定义了图$G$的\emph{acquaintance time}，用$AC(G)$表示，是直到每两个代理都熟悉所需的最小轮数。我们首先研究了一些自然图族的相识时间，包括路、扩张子、二叉树和完全二部图。我们还证明了对所有正整数$n$和$k \leq n^{1.5}$，存在一个$n$ -顶点图$G$使得$AC(G) =Θ(k)$ .我们还证明了，对于所有的$n$ -顶点连通图$G$，我们有$AC(G) = O\left(\frac{n^2}{\log(n)/\log\log(n)}\right)$，提高$O(n^2)$平凡的上限实现顺序地让每个代理执行深度优先搜索沿$G$的生成树。研究了这个问题的计算复杂性，证明了对于任意常数$t \geq 1$，判定给定图$G$具有$AC(G) \leq t$或$AC(G) \geq 2t$的问题是$\mathcal{NP}$ -完全的.也就是说，$AC(G)$是$\mathcal{NP}$ -很难在2的乘法因子内以及在任何加法常数因子内近似。在算法方面，我们给出了一个确定性算法，该算法在给定具有$AC(G)=1$的图$G$的情况下找到了时间上熟人的${\lceil n/c\rceil}$ -回合策略$n^{c+O(1)}$。我们还设计了一个随机多项式时间算法，给出了一个图$G$与$AC(G)=1$发现与高概率的$O(\log(n))$轮策略的熟人。
英文摘要：
We define the following parameter of connected graphs. For a given graph $G$ we place one agent in each vertex of $G$. Every pair of agents sharing a common edge is declared to be acquainted. In each round we choose some matching of $G$ (not necessarily a maximal matching), and for each edge in the matching the agents on this edge swap places. After the swap, again, every pair of agents sharing a common edge become acquainted, and the process continues. We define the \emph{acquaintance time} of a graph $G$, denoted by $AC(G)$, to be the minimal number of rounds required until every two agents are acquainted. We first study the acquaintance time for some natural families of graphs including the path, expanders, the binary tree, and the complete bipartite graph. We also show that for all positive integers $n$ and $k \leq n^{1.5}$ there exists an $n$-vertex graph $G$ such that $AC(G) =Θ(k)$. We also prove that for all $n$-vertex connected graphs $G$ we have $AC(G) = O\left(\frac{n^2}{\log(n)/\log\log(n)}\right)$, improving the $O(n^2)$ trivial upper bound achieved by sequentially letting each agent perform depth-first search along a spanning tree of $G$. Studying the computational complexity of this problem, we prove that for any constant $t \geq 1$ the problem of deciding that a given graph $G$ has $AC(G) \leq t$ or $AC(G) \geq 2t$ is $\mathcal{NP}$-complete. That is, $AC(G)$ is $\mathcal{NP}$-hard to approximate within multiplicative factor of 2, as well as within any additive constant factor. On the algorithmic side, we give a deterministic algorithm that given a graph $G$ with $AC(G)=1$ finds a ${\lceil n/c\rceil}$-rounds strategy for acquaintance in time $n^{c+O(1)}$. We also design a randomized polynomial time algorithm that given a graph $G$ with $AC(G)=1$ finds with high probability an $O(\log(n))$-rounds strategy for acquaintance.
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