The Parameterized Complexity of Welfare Guarantees in Schelling Segregation
Abstract
Schelling's model considers $k$ types of agents each of whom needs to select a vertex on an undirected graph, where every agent prefers to neighbor agents of the same type. We are motivated by a recent line of work that studies solutions that are optimal with respect to notions related to the welfare of the agents. We explore the parameterized complexity of computing such solutions. We focus on the wellstudied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of groupwelfare optimality (GWO) and utilityvector optimality (UVO), both of which lie between WO and PO. Firstly, we focus on the fundamental case where $k=2$ and there are $r$ red agents and $b$ blue agents. We show that all solutionnotions we consider are $\textsf{NP}$hard to compute even when $b=1$ and that they are $\textsf{W}[1]$hard when parameterized by $r$ and $b$. In addition, we show that WO and GWO are $\textsf{NP}$hard even on cubic graphs. We complement these negative results by an $\textsf{FPT}$ algorithm parameterized by $r, b$ and the maximum degree of the graph. For the general case with $k$ types of agents, we prove that for any of the notions we consider the problem is $\textsf{W}[1]$hard when parameterized by $k$ for a large family of graphs that includes trees. We accompany these negative results with an $\textsf{XP}$ algorithm parameterized by $k$ and the treewidth of the graph.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.06904
 Bibcode:
 2022arXiv220106904D
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computer Science and Game Theory;
 91A43;
 91A68
 EPrint:
 11 pages double column