On Reachable Assignments under Dichotomous Preferences

TL;DR

This paper investigates the complexity of reaching a specific item assignment through pairwise exchanges among agents with dichotomous preferences, considering communication constraints modeled by a graph, and establishes complexity results for various graph structures.

Contribution

It proves PSPACE-completeness of the problem even with complete communication graphs and provides polynomial-time algorithms for tree-structured communication graphs.

Findings

PSPACE-complete even with complete graphs

Polynomial-time solution for tree graphs

Complexity depends on the structure of the communication graph

Abstract

We consider the problem of determining whether a target item assignment can be reached from an initial item assignment by a sequence of pairwise exchanges of items between agents. In particular, we consider the situation where each agent has a dichotomous preference over the items, that is, each agent evaluates each item as acceptable or unacceptable. Furthermore, we assume that communication between agents is limited, and the relationship is represented by an undirected graph. Then, a pair of agents can exchange their items only if they are connected by an edge and the involved items are acceptable. We prove that this problem is PSPACE-complete even when the communication graph is complete (that is, every pair of agents can exchange their items), and this problem can be solved in polynomial time if an input graph is a tree.