Resolving the Optimal Metric Distortion Conjecture
Why this work is in the frame
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Bibliographic record
Abstract
We study the following metric distortion problem: there are two finite sets of points, V and C, that lie in the same metric space, and our goal is to choose a point in C whose total distance from the points in V is as small as possible. However, rather than having access to the underlying distance metric, we only know, for each point in V, a ranking of its distances to the points in C. We propose algorithms that choose a point in C using only these rankings as input and we provide bounds on their distortion (worst-case approximation ratio). A prominent motivation for this problem comes from voting theory, where V represents a set of voters, C represents a set of candidates, and the rankings correspond to ordinal preferences of the voters. A major conjecture in this framework is that the optimal deterministic algorithm has distortion 3. We resolve this conjecture by providing a polynomial-time algorithm that achieves distortion 3, matching a known lower bound. We do so by proving a novel lemma about matching voters to candidates, which we refer to as the ranking-matching lemma. This lemma induces a family of novel algorithms, which may be of independent interest, and we show that a special algorithm in this family achieves distortion 3. We also provide more refined, parameterized, bounds using the notion of decisiveness, which quantifies the extent to which a voter may prefer her top choice relative to all others. Finally, we introduce a new randomized algorithm with improved distortion compared to known results, and also provide improved lower bounds on the distortion of all deterministic and randomized algorithms.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it