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arxiv: 2605.04701 · v1 · submitted 2026-05-06 · 💻 cs.GT

Recognition: unknown

When Graph Traversal Meets Structured Preferences: Unified Framework and Complexity Results

Guozhen Rong , Xin Li , Yongjie Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:36 UTC · model grok-4.3

classification 💻 cs.GT
keywords structured preferencesgraph traversalsBFSDFSNP-hardnesspreference profilessupport graphssocial choice
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The pith

Recognizing whether voter preferences come from a graph traversal is NP-hard when the graph has few edges or low degree.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper creates a framework that treats each voter's ranking as the exact order in which candidates are visited when a graph is searched by one of six standard algorithms: BFS, DFS, LexBFS, LexDFS, MCS or MNS. It then studies the recognition problem of deciding, for a given profile, whether any graph exists whose traversals reproduce all the rankings. The authors prove that this decision is NP-hard for every one of the six methods once the support graph is required to have at most k edges, and the same hardness holds when the maximum degree is bounded by k. For DFS the special case of tree supports can be solved in polynomial time, while existing results give polynomial-time algorithms for tree supports under the remaining methods except BFS and LexBFS, whose status is left open.

Core claim

For each of the six traversal paradigms the problem of deciding whether a preference profile admits a support graph with at most k edges is NP-hard; the same holds when the support is required only to have maximum degree k. For the depth-first search paradigm, the existence of a tree support can be decided in polynomial time. Polynomial-time solvability for tree supports under the other five paradigms follows from prior results, except that the complexity for BFS and LexBFS remains open.

What carries the argument

A graph support for a preference profile under a traversal paradigm is a graph on the candidates such that each ranking is exactly the sequence of vertices visited by running the chosen search algorithm on that graph.

If this is right

  • When the support graph is forced to be sparse, no efficient algorithm exists to check whether the profile is consistent with any of the six traversals.
  • Bounded-degree supports do not make the recognition problem easier; hardness persists for every paradigm.
  • For DFS, the existence of a tree support can be verified efficiently, giving a practical test when the underlying structure is acyclic.
  • Polynomial-time methods for tree supports under LexDFS, MCS and MNS follow directly from earlier graph-theoretic results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be used to generate synthetic preference data whose structure is controlled by the choice of traversal and graph size.
  • If real-world preferences often admit tree supports, then efficient checks are available for DFS but may still be missing for BFS.
  • The results suggest studying approximation versions that seek a graph whose traversals come close to the observed rankings rather than matching them exactly.

Load-bearing premise

The observed preferences must be generated exactly by the visit sequences of the chosen graph traversal algorithms.

What would settle it

A concrete family of preference profiles for which the existence of a support graph with at most five edges can be decided in polynomial time under BFS would falsify the NP-hardness claim.

Figures

Figures reproduced from arXiv: 2605.04701 by Guozhen Rong, Xin Li, Yongjie Yang.

Figure 1
Figure 1. Figure 1: Pseudocode for several graph search paradigms. Each algorithm takes as input a graph view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of four-point ordering characterizations of several graph search paradigms. Solid view at source ↗
Figure 4
Figure 4. Figure 4: A (4, 5)-drone. Outline of the Reductions. In several of our reductions, we shall construct a set of orderings that can be classified into two groups. The first group is designed to ensure that any graph with the minimum number of edges (respectively, maximum degree) that admits these orderings as S-orderings contain all edges in (p, q)-drones for some specific values of p and q. This is typically achieved… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the graph G′ constructed in the proof of Theorem 3.7. The three edges between each ux and the dashed boundary indicates that ux is adjacent to all vertices enclosed by the boundary. Vertices belonging to the same set are shown in the same color. In the reduction, we assume that G is 3-regular and contains at least ten vertices. For simplicity, the figure depicts the construction of G′ from … view at source ↗
Figure 6
Figure 6. Figure 6: An illustration of the graph G′ used in the proof of Theorem 3.9. Vertices in the shaded regions represent cliques, while vertices in the white regions form independent sets. Notice that W ∪ F is an independent set in G′ view at source ↗
Figure 7
Figure 7. Figure 7: An illustration of the addition of the six edges (colored in blue) in (2). view at source ↗
Figure 8
Figure 8. Figure 8: An illustration of the ordering σ. Forced vertices are shown in black and free vertices in white. The arc represents the edge in T ⋆ between the forced vertices w ′ and w. The vertex u is the guider, in σ, of all white vertices lying between u and w. Lemma 4.8. Let σ be an ordering of V , let v be an free vertex, and let u = guider(σ, v). Define w as follows: w = ( top(σ), if there is no forced vertex to t… view at source ↗
Figure 9
Figure 9. Figure 9: An illustration of the relative ordering of vertices in the ordering view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops a unified framework modeling voter preferences as traversal orders from six graph search paradigms (BFS, DFS, LexBFS, LexDFS, MCS, MNS) on a common graph. It studies the recognition problem of whether a preference profile admits a graph support, establishing NP-hardness for all paradigms when the support has at most k edges or maximum degree k. It further shows a polynomial-time algorithm for recognizing tree supports under DFS and notes that polynomial-time recognition holds for tree supports under the remaining paradigms except BFS and LexBFS, which remain open.

Significance. If correct, the results meaningfully connect computational social choice with graph algorithms by defining preference restrictions via traversal orders. The NP-hardness results for bounded-edge and bounded-degree supports establish intractability of finding compact realizations, while the polynomial-time results for trees provide algorithmic value. The unified treatment across paradigms and the identification of open problems for BFS and LexBFS are useful for guiding follow-up work. The modeling framework appears internally consistent with standard reductions and prior algorithms.

minor comments (2)
  1. Abstract: the reference to 'existing results' implying polynomial-time solvability for tree supports (except BFS and LexBFS) would be strengthened by citing the specific prior works or theorems being invoked.
  2. The manuscript would benefit from a summary table (perhaps in the introduction or conclusion) listing the complexity status for each paradigm under the edge-bounded, degree-bounded, and tree-support cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; claims rest on independent reductions and algorithms

full rationale

The paper defines a modeling framework that interprets voter preferences as outputs of standard graph traversal algorithms (BFS, DFS, LexBFS, etc.) on a shared graph. It then proves NP-hardness of recognizing supports with at most k edges (or max degree k) via reductions that operate on this model without presupposing the target hardness result. The polynomial-time algorithm for DFS tree supports is shown directly, and other cases are dispatched by citing prior independent results. No step equates a derived quantity to its input by construction, renames a known pattern as a new theorem, or relies on a self-citation chain for the central claims. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard graph theory definitions and complexity theory axioms; no free parameters or invented entities are introduced.

axioms (1)
  • standard math NP-hardness via polynomial-time reductions from known hard problems in complexity theory
    The hardness proofs for bounded edges and degrees rely on this standard technique.

pith-pipeline@v0.9.0 · 5544 in / 1166 out tokens · 50225 ms · 2026-05-08T15:36:05.518135+00:00 · methodology

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