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arxiv: 2509.19279 · v2 · pith:EDHFOJLXnew · submitted 2025-09-23 · 💻 cs.GT

Approximating Electoral Control Problems

Huy Vu Bui , Michael C. Chavrimootoo , Kien T. Le , Son M. Nguyen This is my paper

Pith reviewed 2026-05-18 13:43 UTC · model grok-4.3

classification 💻 cs.GT
keywords electoral controlapproximabilitypluralityapproval votingCondorcetweighted voterscomputational complexityelection attacks
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The pith

Electoral control problems under plurality, approval, and Condorcet are completely classified by their approximability in weighted and unweighted settings.

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

This paper addresses the approximability of electoral control problems, where an attacker modifies the election by adding or deleting candidates or voters to favor a specific winner. While prior research determined the exact computational complexity of these problems, approximability has been less explored despite its relevance to related attacks like manipulation. The authors provide a full classification for the standard control problems using plurality, approval, and Condorcet rules. They establish whether polynomial-time approximation algorithms exist or if the problems are hard to approximate, and they do this for both weighted and unweighted voters.

Core claim

We completely determine 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.

What carries the argument

Standard electoral control problems of adding or deleting candidates or voters to ensure a preferred candidate wins, analyzed via polynomial-time approximation ratios.

If this is right

  • Control problems shown approximable admit efficient algorithms that find near-optimal sets of additions or deletions.
  • Inapproximable problems admit no polynomial-time algorithm with any constant-factor guarantee unless P equals NP.
  • The approximability status is established separately and completely for the weighted-voter case where voters carry different strengths.
  • These optimization results extend earlier exact decision-complexity classifications for the same three voting rules.

Where Pith is reading between the lines

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

  • Election designers could favor rules where more control actions are inapproximable to raise the bar against realistic attacks.
  • The same classification method could be applied to additional rules such as Borda or ranked-choice voting.
  • Practical election-monitoring tools might incorporate the approximable cases as models for likely attacker behavior.
  • Weighted versus unweighted distinctions suggest that influence distribution among voters affects how hard control attacks are to approximate.

Load-bearing premise

The classification assumes the standard control problems are exactly those previously formalized in the literature as adding or deleting candidates or voters, and that approximability uses usual polynomial-time approximation ratio definitions for NP-hard optimization problems.

What would settle it

Discovery of a polynomial-time approximation algorithm for any control problem the paper classifies as inapproximable, or an inapproximability proof for one classified as approximable, under plurality, approval, or Condorcet would disprove the complete determination.

read the original 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. Approximability on the other hand has received little attention in electoral control, despite its prevalence in the study of other forms of electoral attacks, such as manipulation and bribery. Early work established preliminary results about popular voting rules such as plurality, approval, and Condorcet. In this paper, we completely determine 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.

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 / 3 minor

Summary. The paper claims to completely classify the approximability status (in the standard sense of polynomial-time approximation ratios for the corresponding optimization problems) of all standard electoral control variants—constructive and destructive control by adding or deleting candidates or voters—under the plurality, approval, and Condorcet voting rules. The classification is provided for both the weighted and unweighted voter models, with explicit approximation algorithms or gap-preserving reductions supplied for each case.

Significance. If the results hold, this work supplies the first exhaustive map of approximability for electoral control, a gap that has persisted despite extensive decision-complexity results in the literature. The dual treatment of weighted and unweighted settings, together with the use of standard approximation-ratio definitions and gap-preserving reductions, makes the classification directly usable for both theoretical follow-up and practical algorithm design in computational social choice.

minor comments (3)
  1. §3, Definition 2: the formalization of the optimization version of control by deleting voters should explicitly state whether the objective is to minimize the number of deletions or to maximize the margin of victory; the current wording leaves this ambiguous for the approximation guarantee.
  2. Table 1: the row for 'Condorcet, constructive, add candidates, weighted' lists 'APX-hard' without citing the specific reduction or theorem number that establishes the inapproximability gap.
  3. §5.2, Algorithm 1: the running-time analysis claims O(n^3) but the pseudocode contains an implicit enumeration over subsets whose size is bounded only by the parameter; clarify whether this is FPT or merely polynomial for fixed weights.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our results and for recommending minor revision. We are pleased that the exhaustive approximability classification for electoral control under plurality, approval, and Condorcet rules (in both weighted and unweighted settings) is viewed as filling a longstanding gap in the literature.

Circularity Check

0 steps flagged

No significant circularity; results rest on independent reductions and prior definitions

full rationale

The manuscript classifies approximability for each standard electoral control variant (constructive/destructive, add/delete candidates/voters) under plurality, approval, and Condorcet, in weighted and unweighted settings. It supplies explicit polynomial-time approximation algorithms or gap-preserving reductions for each case, drawing on established NP-hardness results and standard approximation-ratio definitions from the literature. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional equivalence; each approximability status is derived separately via case analysis and external complexity tools. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard definitions of electoral control problems and approximation ratios from prior literature in computational social choice; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Standard definitions of control actions (add/delete candidates/voters) and voting rules (plurality, approval, Condorcet) as used in prior work.
    Invoked when stating the problems whose approximability is classified.
  • standard math Polynomial-time approximation ratio is the appropriate measure for these optimization problems.
    Underlying the decision of whether each problem is approximable.

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discussion (0)

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Reference graph

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