Exact capacitated domination: on the computational complexity of uniqueness

Abstract

We explore the uniqueness of pure strategy Nash equilibria in the Netflix Games of Gerke et al. (arXiv:1905.01693, 2019. Let G = (V, E) be a graph and κ : V → Z ≥0 a function, and call the pair (G, κ) a capacitated graph. A spanning subgraph H of (G, κ) is called a DP -Nash subgraph if H is bipartite with partite sets X, Y called the D-set and P -set of H, respectively, such that no vertex of P is isolated and for every x ∈ X, dH (x) = min{dG(x), κ(x)}. We prove that whether (G, κ) has a unique DP -Nash subgraph can be decided in polynomial time. We also show that when κ(v) = k for every v ∈ V , the problem of deciding whether (G, κ) has a unique D-set is polynomial time solvable for k = 0 and 1, and co-NP-complete for k ≥ 2.

The Effect of Social Distancing on the Reach of an Epidemic in Social Networks

“…We do not mean individuals maintaining a predetermined physical distance throughout an encounter. Our measure is adapted from the gametheoretic model of Netflix Games (Gerke et al, 2019;Gutin et al, 2020). There, the purchaser of a product may only share with a limited number of friends.…”