The physics of custody

TL;DR

This paper models complex custody arrangements among divorced individuals and their new partners as a spin glass system, revealing the problem's NP-Complete nature and providing insights into maximizing compatible couple groupings.

Contribution

It formulates custody arrangement challenges as a spin glass and max-cut problem, demonstrating the computational complexity and offering a novel theoretical perspective.

Findings

Custody arrangement problem is NP-Complete.

Maximizing compatible couple groupings is computationally hard.

Modeling custody as a spin glass provides new analytical tools.

Abstract

Divorced individuals face complex situations when they have children with different ex-partners, or even more, when their new partners have children of their own. In such cases, and when kids spend every other weekend with each parent, a practical problem emerges: Is it possible to have such a custody arrangement that every couple has either all of the kids together or no kids at all? We show that in general, it is not possible, but that the number of couples that do can be maximized. The problem turns out to be equivalent to finding the ground state of a spin glass system, which is known to be equivalent to what is called a weighted max-cut problem in graph theory, and hence it is NP-Complete.