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Bi-Criteria Metric Distortion

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Social Choice TheoryMetric Distortion
Abstract:

Selecting representatives based on voters' preferences is a fundamental problem in social choice theory. While cardinal utility functions offer a detailed representation of preferences, voters often cannot precisely quantify their affinity towards a given candidate. As a result, modern voting systems rely on ordinal rankings to simplistically represent preference profiles. In quantifying the suboptimality of solutions due to the loss of information when using ordinal preferences, the metric distortion framework models voters and candidates as points in a metric space, with distortion bounding the efficiency loss. Prior works within this framework use the distance between a voter and a candidate in the underlying metric as the cost of selecting the candidate for the given voter, with a goal of minimizing the sum (utilitarian) or maximum (egalitarian) of costs across voters. For deterministic election mechanisms selecting a single winning candidate, the best possible distortion is known to be 3 for any metric, as established by Gkatzelis, Halpern, and Shah (FOCS'20). In contrast, for randomized mechanisms, distortions cannot be lower than 2.1122.112, as shown by Charikar and Ramakrishnan (SODA'22), and there exists a mechanism with a distortion guarantee of 2.7532.753, according to Charikar, Ramakrishnan, Wang, and Wu (SODA'24 Best Paper Award). Our work asks: can one obtain a better approximation compared to an optimal candidate by selecting a committee of kk candidates (k≥1k \ge 1), where the cost of a voter is defined to be its distance to the closest candidate in the committee? We affirmatively answer this question by introducing the concept of bi-criteria approximation within the metric distortion framework. In the line metric, it is possible to achieve optimal cost with only O(1)O(1) candidates. In contrast, we also prove that in both the two-dimensional and tree metrics -- which naturally generalize the line metric -- achieving optimal cost is impossible unless all candidates are selected. These results apply to both utilitarian and egalitarian objectives. Our results establish a stark separation between the line metric and the 2D or tree metric in the context of the metric distortion problem.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper introduces a bi-criteria approximation framework for metric distortion in committee selection, where voters and candidates lie in an unknown metric space and only ordinal rankings are observed. It sits in the 'Bi-Criteria and k-Committee Distortion Analysis' leaf, which contains just two papers total. This leaf is part of the broader 'General Multi-Winner Distortion Bounds and Mechanisms' branch, indicating a relatively sparse research direction focused on constant-factor approximations for k-committee problems without explicit fairness constraints.

The taxonomy reveals that committee selection research divides into proportionality-focused work (three papers on fairness and representation) and general multi-winner bounds (seven papers across three leaves). The original paper's leaf neighbors 'Multi-Winner Distortion in Specific Metric Spaces' (two papers on line metrics) and 'Multi-Winner Voting with Single-Candidate Ballots' (one paper). Its sibling paper, Constant-Factor k-Committee, shares the focus on tight multi-winner bounds but does not explicitly explore bi-criteria trade-offs. Nearby branches address single-winner optimal distortion (seven papers) and distributed mechanisms (three papers), suggesting the paper bridges multi-winner theory with optimization techniques from single-winner settings.

Among thirteen candidates examined, no contribution was clearly refuted. The bi-criteria framework (three candidates examined, zero refutable) appears novel within the limited search scope, as does the optimal cost result for line metrics (three candidates, zero refutable). The separation between line and higher-dimensional metrics (seven candidates, zero refutable) also shows no direct prior work among the examined papers. However, the search scope is modest: thirteen candidates total, not hundreds, so these findings reflect top-K semantic matches and citation expansion rather than exhaustive coverage of the field.

Given the sparse taxonomy leaf and absence of refuting prior work among examined candidates, the contributions appear relatively novel within the limited search context. The bi-criteria relaxation offers a fresh angle on multi-winner distortion, though the small candidate pool means potentially relevant work outside the top-K matches may exist. The analysis captures the paper's position in an active but not overcrowded research direction, with room for further exploration of fairness-distortion trade-offs.

Taxonomy

Core-task Taxonomy Papers
20
3
Claimed Contributions
13
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: bi-criteria approximation in metric distortion for committee selection. The field of metric distortion in voting seeks to design mechanisms that perform well when voters' true preferences are distances in an unknown metric space, but the mechanism only observes ordinal rankings. The taxonomy reflects a natural division into four main branches: Committee Selection and Multi-Winner Voting, which studies how to choose multiple winners while bounding worst-case distortion; Single-Winner Voting and Optimal Distortion, focusing on tight bounds and optimal rules for selecting a single alternative; Distributed Voting Mechanisms, examining settings where voters are partitioned across multiple districts or agents; and Applications and Extensions Beyond Voting, exploring connections to facility location, clustering, and other domains. Within the committee selection branch, research ranges from general multi-winner bounds to specialized analyses of proportional representation and fairness-aware mechanisms, with works like Multiwinner Voting Distortion[6] and Proportional Committee Selection[4] establishing foundational results. Recent progress has concentrated on refining distortion guarantees for k-committee problems, where the challenge is to balance global social cost with fairness or proportionality constraints. A handful of studies, including Constant-Factor k-Committee[1] and Bi-Criteria Metric Distortion[0], push toward constant-factor approximations by leveraging bi-criteria frameworks that relax one objective to improve another. Bi-Criteria Metric Distortion[0] sits squarely within this active line, closely aligned with Constant-Factor k-Committee[1] in its focus on tight multi-winner bounds, yet it explores trade-offs between different notions of distortion rather than a single worst-case measure. Meanwhile, neighboring work such as Proportional Fairness Clustering[5] and Breaking Metric Barrier[3] highlights ongoing debates about whether proportionality, fairness, or pure utilitarian distortion should take precedence, and whether deterministic or randomized mechanisms offer better guarantees. The original paper contributes to this landscape by formalizing bi-criteria relaxations that may enable improved approximation ratios without sacrificing all fairness properties.

Claimed Contributions

Bi-criteria approximation framework for metric distortion

The authors introduce a bi-criteria approximation perspective to the metric distortion problem, where a committee of k candidates is selected and compared against the cost of a single optimal candidate. This framework allows achieving better approximations than the classical single-winner distortion bounds.

3 retrieved papers
Optimal cost with constant-size committees in line metric

For the line metric, the authors demonstrate that a constant number of candidates suffices to match the cost of an optimal single candidate. Specifically, two candidates achieve optimal cost for the utilitarian objective, and four candidates achieve optimal cost for the egalitarian objective.

3 retrieved papers
Separation between line metric and higher-dimensional metrics

The authors prove that achieving optimal cost in 2D Euclidean or tree metrics is impossible unless all candidates are selected, contrasting sharply with the line metric where constant-size committees suffice. This establishes a fundamental structural difference between these metric spaces in the metric distortion framework.

7 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Bi-criteria approximation framework for metric distortion

The authors introduce a bi-criteria approximation perspective to the metric distortion problem, where a committee of k candidates is selected and compared against the cost of a single optimal candidate. This framework allows achieving better approximations than the classical single-winner distortion bounds.

Contribution

Optimal cost with constant-size committees in line metric

For the line metric, the authors demonstrate that a constant number of candidates suffices to match the cost of an optimal single candidate. Specifically, two candidates achieve optimal cost for the utilitarian objective, and four candidates achieve optimal cost for the egalitarian objective.

Contribution

Separation between line metric and higher-dimensional metrics

The authors prove that achieving optimal cost in 2D Euclidean or tree metrics is impossible unless all candidates are selected, contrasting sharply with the line metric where constant-size committees suffice. This establishes a fundamental structural difference between these metric spaces in the metric distortion framework.

Table of Contents

Bi-Criteria Metric Distortion | Novelty Validation