How Expressive Are Friendly School Partitions?
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≥ 3 there always exist at least 2 friendly partitions and when d≥ 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.READ FULL TEXT