Fair subgraph selection for contagion containment (Brief Announcement)

We present a new class of problems where the goal is to select a "fair" subgraph H of a given graph G = (V,E), such that H decomposes into many small components. A subgraph H c G is (P,d) fair if every vertex v ϵ P has the same degree d in H, where P c V and d > 0 are input parameters. These problems arise when the goal is to allow individuals to equally participate in activities in such a way that the connected components within an interaction graph, which models potential interactions among people, are of the smallest possible size, so that the spread of the contagion, and the difficulty of contact tracing in case of infection, is minimized. Within a preference graph that models the set of preferred choices for each individual when selecting among available options of where to conduct any particular type of activity (e.g., which gym to attend), we seek to compute the fair subgraph of assignments of individuals to these options, so that the number of people in each connected component ("interaction community") of the resulting subgraph is minimized, and everyone is given the same number of options for every activity. We show that the fair subgraph selection problem is NP-hard, even for very restricted versions. We then formulate the problem as an integer program, and give a polynomial time computable lower bound on the optimal solution.