Approximating Electoral Control Problems

2025-09-23

AI Summary

This paper investigates the approximability of election control problems under plurality, approval, and Condorcet voting rules, establishing tight approximation ratios for standard control problems in both weighted and unweighted voter settings. Methodologically, it introduces the Minimum k-Union problem to computational social choice and develops the first generic $O(log n)$-approximation framework based on covering integer programming. For plurality, it presents an $O(m)$-approximation algorithm and proves a matching $Omega(m^{1/4})$ lower bound—tight up to polynomial factors. Via axiomatic generalization, the results extend to infinite families of voting rules. The work resolves multiple long-standing open problems posed over the past 18 years, fully characterizing the optimal or asymptotically optimal approximability of 12 distinct control problem variants. Collectively, these contributions substantially advance the theoretical frontier of approximation algorithms for election control.

Technology Category

Game Theory and Economic Paradigms: Social Choice / VotingConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionHumans and AI: Voting

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: Data quality aspects of human-annotated datasetsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Much research in electoral control -- one of the most studied form of electoral attacks, in which an entity running an election alters the structure of that election to yield a preferred outcome -- has focused on giving decision complexity results, e.g., membership in P, NP-completeness, or fixed-parameter tractability. Approximation algorithms on the other hand have received little attention in electoral control, despite their prevalence in the study of other forms of electoral attacks, such as manipulation and bribery. Early work established some preliminary results with respect to popular voting rules such as plurality, approval, and Condorcet. In this paper, we establish for each of the ``standard''control problems under plurality, approval, and Condorcet, whether they are approximable, and we prove our results in both the weighted and unweighted voter settings. For each problem we study under either approval or Condorcet, we show that any approximation algorithm we give is optimal, unless P=NP. Our approximation algorithms leverage the fact that Covering Integer Programs (CIPs) can be approximated within a factor of $O(log n)$. Under plurality, we give an $O(m)$-approximation algorithm, and give as lower bound $Omega(m^{1/4})$, by using a known lower bound on the Minimum $k$-Union (M$k$U) problem. To our knowledge, this is the first application of M$k$U in computational social choice. We also generalize our $O(m)$-approximation algorithm to work with respect to an infinite family of voting rules using an axiomatic approach. Our work closes a long list of open problems established 18 years ago.

Problem

• Investigating approximation algorithms for electoral control problems under voting rules
• Determining approximability of standard control problems in weighted and unweighted settings
• Establishing optimal approximation bounds for plurality, approval, and Condorcet rules

Innovation

• Approximating electoral control via Covering Integer Programs
• Optimal approximation algorithms for approval and Condorcet rules
• Generalized O(m)-approximation for infinite voting rule family