The Family Holiday Gathering Problem or Fair and Periodic Scheduling of Independent Sets

TL;DR

This paper introduces the Holiday Gathering Problem, modeling family holiday scheduling as a graph problem, and proposes periodic, coloring-based solutions with optimal bounds, connecting it to classical scheduling and encoding problems.

Contribution

It formulates the Holiday Gathering Problem, establishes optimal periodic scheduling solutions using coloring and prefix-free encodings, and links it to classical problems in computer science.

Findings

Coloring-based solutions with periods depending on node color and degree.

Optimality of the proposed coloring-based solutions.

Construction of periodic schedules with period at most 2d for degree d nodes.

Abstract

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 $d$.