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.

翻译：偏好限制在计算社会选择中发挥了重要作用。本文研究了一个将偏好限制与经典图搜索范式相连接的框架。我们将候选人建模为图的顶点，并将每位选民的偏好排序解释为根据某种图搜索遍历图的结果。我们关注六种基本范式：广度优先搜索（BFS）、深度优先搜索（DFS）、字典序广度优先搜索（LexBFS）、字典序深度优先搜索（LexDFS）、最大基数搜索（MCS）和最大邻域搜索（MNS）。在此框架内，我们研究确定给定偏好分布是否在结构限制下允许图支持的问题，即是否存在一个图，使得每种偏好排序可以通过所选范式遍历该图而生成。对于所有考虑的范式，我们证明当要求图支持最多有$k$条边（其中$k$是给定整数）时，该问题是NP难的。我们进一步将这些困难性结果扩展到要求图支持的最大度数为$k$的情况。对于DFS，我们证明识别偏好分布是否允许树支持可以在多项式时间内求解。此外，现有结果表明，除BFS和LexBFS外，所有其他图遍历问题的多项式时间可解性都成立，而BFS和LexBFS的复杂性仍悬而未决。