Bi-Criteria Metric Distortion
Kiarash Banihashem, Diptarka Chakraborty, Shayan Chashm Jahan, Iman, Gholami, MohammadTaghi Hajiaghayi, Mohammad Mahdavi, Max Springer
TL;DR
This paper investigates how selecting a small committee of candidates in a metric space can significantly reduce social choice distortion, achieving optimal costs on line metrics and providing bounds for other metric spaces.
Contribution
It demonstrates that a small committee can outperform single-candidate solutions in line metrics, offering new upper bounds and lower bounds for metric distortion.
Findings
Optimal cost achieved with O(1) candidates on line metrics
Extends results to utilitarian and egalitarian objectives
Provides lower bounds for line and 2-D Euclidean metrics
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, ordinal rankings are often the only available information due to their simplicity and practical constraints. The metric distortion framework addresses this issue by modeling voters and candidates as points in a metric space, with distortion quantifying the efficiency loss from relying solely on ordinal rankings. Existing works define the cost of a voter with respect to a candidate as their distance and set the overall cost as either the sum (utilitarian) or maximum (egalitarian) of these costs across all voters. They show that deterministic algorithms achieve a best-possible distortion of 3 for any metric when considering a single candidate. This paper explores whether one can obtain a better…
Peer Reviews
Decision·ICLR 2026 Poster
The paper studies a very natural question: how many additional candidates are needed to guarantee optimal distortion. This to some extend related with k-committee selection, but the main question posed here is different. The paper obtains strong and non-trivial results both both the utilitarian and egalitarian objectives, and both the 1D case and more general metrics. The results on the line are particularly surprising: a constant number of candidates (and in fact a small constant at that) alway
The main weakness is that, a priori, selecting more than one winning candidate could trivialize the problem when comparing against the optimal omniscient solution that picks a single candidate. Still, the lower bounds of the paper show that this is far from trivial in general, and accomplishing it in 1D turns out to be highly non-trivial.
The paper introduces a bi-criteria approximation framework to analyze how selecting multiple candidates can reduce efficiency loss in metric distortion, extending classical single-winner results to multi-candidate settings for both utilitarian and egalitarian objectives. It establishes precise trade-offs between committee size and approximation quality, proving that on the line metric two candidates can achieve optimal cost, while in higher-dimensional and tree metrics such improvement is imposs
The main weakness of the paper is in the presentation, given that the introduction is too long and most proofs of correctness are relegated to the appendix, which is pretty long. On the other hand, the results are presented well and given that the paper is technically demanding, this choice is justified.
* I think the content of the paper is good, and I feel that between the appendices and the main body a strong paper can be found (more in weaknesses/suggestions) * The content is well motivated, I believe it is a novel contribution, and well-justified.
* I thought the visuals in the paper were quite weak and to the detriment of the clarity of the paper. There are only 2 figures. The first is quite large but has conveys relatively little information. The next (2/3) one doesn't convey much beyond the writing. I felt that understandability could be easily enhanced with more visuals that convey more meaning (there are several in the appendix that seem to be much better/efficient at conveying information) * Perhaps the above could be overcome, but
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Taxonomy
TopicsAdvanced Measurement and Metrology Techniques