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The Strong Maximum Circulation Algorithm: A New Method for Aggregating Preference Rankings

Nathan Atkinson, Scott C. Ganz, Dorit S. Hochbaum, James B. Orlin

TL;DR

This work introduces the strong maximum circulation framework to aggregate pairwise preferences by removing maximal circulations from a vote graph, treating cycles as ties to derive a unique strong partial order. The core algorithm (Perturbation) solves a single maximum-cost flow problem to obtain a strong maximum circulation and certifying dual solution, ensuring a unique consensus. It also connects the approach to established optimization models via the Hochbaum–Levin framework, showing that the dual of the maximum circulation problem corresponds to a convex relaxation of Kemeny’s method. The paper further analyzes the computational limits by studying minimum maximal circulations, proving NP-hardness and illustrating that such eliminations can yield conflicting partial orders. Overall, the strong maximum circulation method provides an efficient, normative alternative to Kemeny with a principled dual relationship to convex ranking objectives, and it opens avenues for exploring other cycle-removal strategies and their empirical performance.

Abstract

We present a new optimization-based method for aggregating preferences in settings where each voter expresses preferences over pairs of alternatives. Our approach to identifying a consensus partial order is motivated by the observation that collections of votes that form a cycle can be treated as collective ties. Our approach then removes unions of cycles of votes, or circulations, from the vote graph and determines aggregate preferences from the remainder. Specifically, we study the removal of maximal circulations attained by any union of cycles the removal of which leaves an acyclic graph. We introduce the strong maximum circulation, the removal of which guarantees a unique outcome in terms of the induced partial order, called the strong partial order. The strong maximum circulation also satisfies strong complementary slackness conditions, and is shown to be solved efficiently as a network flow problem. We further establish the relationship between the dual of the maximum circulation problem and Kemeny's method, a popular optimization-based approach for preference aggregation. We also show that identifying a minimum maximal circulation -- i.e., a maximal circulation containing the smallest number of votes -- is an NP-hard problem. Further an instance of the minimum maximal circulation may have multiple optimal solutions whose removal results in conflicting partial orders.

The Strong Maximum Circulation Algorithm: A New Method for Aggregating Preference Rankings

TL;DR

This work introduces the strong maximum circulation framework to aggregate pairwise preferences by removing maximal circulations from a vote graph, treating cycles as ties to derive a unique strong partial order. The core algorithm (Perturbation) solves a single maximum-cost flow problem to obtain a strong maximum circulation and certifying dual solution, ensuring a unique consensus. It also connects the approach to established optimization models via the Hochbaum–Levin framework, showing that the dual of the maximum circulation problem corresponds to a convex relaxation of Kemeny’s method. The paper further analyzes the computational limits by studying minimum maximal circulations, proving NP-hardness and illustrating that such eliminations can yield conflicting partial orders. Overall, the strong maximum circulation method provides an efficient, normative alternative to Kemeny with a principled dual relationship to convex ranking objectives, and it opens avenues for exploring other cycle-removal strategies and their empirical performance.

Abstract

We present a new optimization-based method for aggregating preferences in settings where each voter expresses preferences over pairs of alternatives. Our approach to identifying a consensus partial order is motivated by the observation that collections of votes that form a cycle can be treated as collective ties. Our approach then removes unions of cycles of votes, or circulations, from the vote graph and determines aggregate preferences from the remainder. Specifically, we study the removal of maximal circulations attained by any union of cycles the removal of which leaves an acyclic graph. We introduce the strong maximum circulation, the removal of which guarantees a unique outcome in terms of the induced partial order, called the strong partial order. The strong maximum circulation also satisfies strong complementary slackness conditions, and is shown to be solved efficiently as a network flow problem. We further establish the relationship between the dual of the maximum circulation problem and Kemeny's method, a popular optimization-based approach for preference aggregation. We also show that identifying a minimum maximal circulation -- i.e., a maximal circulation containing the smallest number of votes -- is an NP-hard problem. Further an instance of the minimum maximal circulation may have multiple optimal solutions whose removal results in conflicting partial orders.
Paper Structure (14 sections, 12 theorems, 17 equations, 2 figures)

This paper contains 14 sections, 12 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Maximum Circulations and Strong Maximum Circulations
  4. Generating a strong maximum circulation based on the acyclic set $A^*$
  5. Constructing a Strong Maximum Circulation
  6. An Illustrative Example
  7. Computing a Strong Circulation with One Maximum Circulation Flow
  8. Strong (Rather than Strict) Complementary Slackness
  9. The "Perturbation" Algorithm
  10. Dual Optimal Rankings and the Hochbaum-Levin Model
  11. Minimum Maximal Circulations
  12. Minimum Maximal Circulations Can Induce Conflicting Partial Orders
  13. Computational Complexity of Finding a Minimum Maximal Circulation
  14. Conclusion

Key Result

Lemma 1

If the vote graph $G = (V, A(\mathbf{q}))$ is Eulerian, then the flow $\mathbf{x}^* = \mathbf{q}$ is the unique maximum circulation, and $A^* = \emptyset$. If the vote graph is not Eulerian, then $\mathbf{x}^* = \mathbf{q}$ is not a circulation, and $A^* \neq \emptyset$.

Figures (2)

  • Figure 1: The representation of the vote graph as a: (a) multi-graph, (b) capacitated graph.
  • Figure 2: Example of a vote graph where minimum maximal circulations induce conflicting partial orders.

Theorems & Definitions (20)

  • Lemma 1
  • Corollary 1
  • Lemma 2
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Theorem 1
  • proof
  • ...and 10 more