How Expressive Are Friendly School Partitions?

TL;DR

This paper investigates the complexity and properties of 'friendly' student partitions in school class assignments, analyzing their existence, quantity, and separability under various constraints.

Contribution

It establishes existence results for friendly partitions and student separability, and explores the limits of their expressiveness under different conditions.

Findings

At least two friendly partitions exist when each student lists at least 3 peers.

A student can be separated from any other when each lists at least 15 peers.

Open questions remain about pairwise separability and conditions for universal separation.

Abstract

A natural procedure for assigning students to classes in the beginning of the school-year is to let each student write down a list of $d$ other students with whom she/he wants to be in the same class (typically $d=3$). The teachers then gather all the lists and try to assign the students to classes in a way that each student is assigned to the same class with at least one student from her/his list. We refer to such partitions as friendly. In realistic scenarios, the teachers may also consider other constraints when picking the friendly partition: e.g. there may be a group of students whom the teachers wish to avoid assigning to the same class; alternatively, there may be two close friends whom the teachers want to put together; etc. Inspired by such challenges, we explore questions concerning the expressiveness of friendly partitions. For example: Does there always exist a friendly partition? More generally, how many friendly partitions are there? Can every student $u$ be separated from any other student $v$? Does there exist a student $u$ that can be separated from any other student $v$? We show that when $d\geq 3$ there always exist at least $2$ friendly partitions and when $d\geq 15$ there always exists a student $u$ which can be separated from any other student $v$. The question regarding separability of each pair of students is left open, but we give a positive answer under the additional assumption that each student appears in at most roughly $\exp(d)$ lists. We further suggest several open questions and present some preliminary findings towards resolving them.