How Expressive Are Friendly School Partitions?
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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.
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