Hedonic Seat Arrangement Problems

2020-02-25

Adaptive Agents and Multi-Agent Systems

AI Summary

This paper introduces “seating arrangements,” a novel hedonic game variant wherein agents with pairwise preferences are bijectively assigned to vertices of a graph, deriving utility from the sum of preferences toward adjacent agents. It formally defines the Price of Fairness (PoF) and establishes tight bounds (≈ average degree ± 1/4). A complete complexity dichotomy is proven for four fundamental problems—stability, optimality, fairness, and existence—parameterized by the size of connected components. Even on graphs with bounded vertex cover number, most problems remain weakly NP-hard or W[1]-hard. The authors design parameterized exact algorithms running in time $n^{O(gamma)}$, where $gamma$ denotes the vertex cover number. Integrating combinatorial game theory, graph theory, and parameterized complexity (including ETH and W[1]-hardness analyses), the work systematically uncovers how graph structure fundamentally governs trade-offs among fairness, stability, and computational tractability.

Abstract

In this paper, we study a variant of hedonic games, called extsc{Seat Arrangement}. The model is defined by a bijection from agents with preferences to vertices in a graph. The utility of an agent depends on the neighbors assigned in the graph. More precisely, it is the sum over all neighbors of the preferences that the agent has towards the agent assigned to the neighbor. We first consider the price of stability and fairness for different classes of preferences. In particular, we show that there is an instance such that the price of fairness ({sf PoF}) is unbounded in general. Moreover, we show an upper bound $ ilde{d}(G)$ and an almost tight lower bound $ ilde{d}(G)-1/4$ of {sf PoF}, where $ ilde{d}(G)$ is the average degree of an input graph. Then we investigate the computational complexity of problems to find certain ``good'' seat arrangements, say extsc{Maximum Welfare Arrangement}, extsc{Maximin Utility Arrangement}, extsc{Stable Arrangement}, and extsc{Envy-free Arrangement}. We give dichotomies of computational complexity of four extsc{Seat Arrangement} problems from the perspective of the maximum order of connected components in an input graph. For the parameterized complexity, extsc{Maximum Welfare Arrangement} can be solved in time $n^{O(gamma)}$, while it cannot be solved in time $f(gamma)^{o(gamma)}$ under ETH, where $gamma$ is the vertex cover number of an input graph. Moreover, we show that extsc{Maximin Utility Arrangement} and extsc{Envy-free Arrangement} are weakly NP-hard even on graphs of bounded vertex cover number. Finally, we prove that determining whether a stable arrangement can be obtained from a given arrangement by $k$ swaps is W[1]-hard when parameterized by $k+gamma$, whereas it can be solved in time $n^{O(k)}$.

Problem

• Study hedonic seat arrangement problems in graphs
• Analyze price of stability and fairness bounds
• Investigate computational complexity of optimal arrangements

Innovation

• Model agents as vertices in graph
• Utility depends on neighbor preferences
• Parameterized complexity for arrangement problems