When Graph Traversal Meets Structured Preferences: Unified Framework and Complexity Results
For researchers in computational social choice and graph theory, this work provides a unified perspective and complexity results, though many results are hardness proofs with open cases remaining.
This paper introduces a framework connecting preference restrictions in computational social choice with graph traversal paradigms, and shows that determining whether a preference profile admits a graph support with at most k edges or maximum degree k is NP-hard for six fundamental graph searches. For DFS, recognizing tree support is polynomial-time solvable.
Preference restrictions have played a significant role in computational social choice. This paper studies a framework that connects preference restrictions with classical graph search paradigms. We model candidates as vertices of a graph and interpret the preference ordering of each voter as the outcome of traversing the graph according to a graph search. We focus on six fundamental paradigms: breadth-first search (BFS), depth-first search (DFS), breadth-first search (LexBFS), lexicographic depth-first (LexDFS), maximum cardinality search (MCS), and maximal neighborhood search (MNS). Within this framework, we study the problem of determining whether a given preference profile admits a graph support subject to structural restrictions, that is, whether there exists a graph such that each preference ordering can be generated by traversing the graph under the chosen paradigm. For all considered paradigms, we show that this problem is NP-hard when the graph support is required to have at most $k$ edges, where $k$ is a given integer. We further extend these hardness results to the case where the graph support is required to have maximum degree $k$. For DFS, we prove that recognizing whether a preference profile admits a tree support can be solved in polynomial time. Moreover, existing results imply polynomial-time solvability of the problem for all remaining graph traversals, except BFS and LexBFS, for which the complexity remains open.