独立集的公平周期排序的家庭假日聚会问题
The Family Holiday Gathering Problem or Fair and Periodic Scheduling of Independent Sets
摘要：
我们介绍和研究{ \em假日聚会问题}模型的困难，夫妇试图决定与父母，他们应该度过假期。我们的目标是安排家庭聚会，让父母{ \em快乐}，即他们所有的孩子都将{ \em同时}回家参加节日庆祝活动，同时最大限度地减少父母不高兴的连续假期的数量。假日聚会问题与计算机科学中的几个经典问题密切相关，例如一般图上的{ \em用餐哲学家问题}和周期调度，并且在蜂窝无线电的传输调度中有应用。我们还展示了周期性调度，着色和通用前缀自由编码之间有趣的联系。假日聚会问题的组合定义是：给定一个图$G$，找出$G$的独立集的无限序列。目标函数是对于每个节点$v$，最小化$v$的两次出现之间的最大间隙。在良好的解决方案中，该间隙取决于节点的局部属性（即，它的度）和解决方案应该是周期性的，即每固定数量的周期出现一个节点。我们展示了一个基于着色的构造，其中用$c$着色的每个节点的周期至多为$2^{1+\log^*c}\cdot\prod_{i=0}^{\log^*c} \log^{(i)}c$（其中$\log^{(i)}$意味着迭代$\log$函数$i$次）。这通过与{ \it无前缀编码}的连接来实现。我们证明，这是最好的基于颜色的解决方案。我们还显示了一个建设周期最多$2d$度$d$的节点。
英文摘要：
We introduce and examine the {\em Holiday Gathering Problem} which models the difficulty that couples have when trying to decide with which parents should they spend the holiday. Our goal is to schedule the family gatherings so that the parents that will be {\em happy}, i.e.\ all their children will be home {\em simultaneously} for the holiday festivities, while minimizing the number of consecutive holidays in which parents are not happy. The holiday gathering problem is closely related to several classical problems in computer science, such as the {\em dining philosophers problem} on a general graph and periodic scheduling,and has applications in scheduling of transmissions made by cellular radios. We also show interesting connections between periodic scheduling, coloring, and universal prefix free encodings. The combinatorial definition of the Holiday Gathering Problem is: given a graph $G$, find an infinite sequence of independent-sets of $G$. The objective function is to minimize, for every node $v$, the maximal gap between two appearances of $v$. In good solutions this gap depends on local properties of the node (i.e., its degree) and the the solution should be periodic, i.e.\ a node appears every fixed number of periods. We show a coloring-based construction where the period of each node colored with the $c$ is at most $2^{1+\log^*c}\cdot\prod_{i=0}^{\log^*c} \log^{(i)}c$ (where $\log^{(i)}$ means iterating the $\log$ function $i$ times). This is achieved via a connection with {\it prefix-free encodings}. We prove that this is the best possible for coloring-based solutions. We also show a construction with period at most $2d$ for a node of degree $d$.
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