The family holiday gathering problem or fair and periodic scheduling of independent sets

We introduce the Holiday Gathering Problem which models the difficulty in scheduling non-interfering transmissions in (wireless) networks. Our goal is to schedule transmission rounds so that the antennas that transmit in a given round will not interfere with each other, i.e. all of the other antennas that can interfere will not transmit in that round, while minimizing the number of consecutive rounds in which antennas do not transmit. Following a long tradition in Computer Science, we introduce the problem by an intuitive story. Assume we live in a perfect world where families enjoy being together. Consequently, parents, whose children are in a monogamous relation, would like to have all their children at home for the holiday meal (i.e. there is a special pleasure gained by hosting all the children simultaneously and they wish to have this event occur as frequently as possible). However, the conflict is that the in-laws would also be happiest if all their children come to them. Our goal can be described as scheduling an infinite sequence of "guest lists" in a distributed setting so that each child knows where it will spend the holiday. The holiday gathering problem is closely related to several classical problems in computer science, such as the dining philosophers problem on a general graph and periodic scheduling. The process of the scheduling should be done with no further communication after initialization, by using a small amount of local data. The result should minimize the number of consecutive holidays where the family is not together. In a good sequence this number depends on local properties of the parents (e.g., their number of children). Furthermore, solutions that are periodic, i.e. a gathering occurs every fixed number of rounds, are useful for maintaining a small amount of information at each node and reducing the amount of ongoing communication and computation. Our algorithmic techniques show interesting connections between periodic scheduling, coloring, and universal prefix free encodings. We develop a coloring-based construction where the period of each node colored with the c is at most 2 +^log∗c.Π_i=0^log∗c log⁽ⁱ⁾ c (where log⁽ⁱ⁾ means iterating the log function i times). This is achieved via a connection with 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.