The Parameterized Complexity of Welfare Guarantees in Schelling Segregation

TL;DR

This paper investigates the computational complexity of finding welfare-optimizing solutions in Schelling segregation models, revealing NP-hardness and W[1]-hardness results, alongside fixed-parameter tractable algorithms under certain parameters.

Contribution

It provides a comprehensive complexity analysis of welfare-based solution concepts in Schelling segregation, including new hardness results and algorithms parameterized by various factors.

Findings

All solution notions are NP-hard to compute even with minimal agents.

They are W[1]-hard when parameterized by the number of red and blue agents.

An FPT algorithm exists when parameterized by agents and graph degree.

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 well-studied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of group-welfare optimality (GWO) and utility-vector 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 solution-notions 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.