Pith. sign in

REVIEW 4 major objections 5 minor 31 references

Bi-Criteria Metric Distortion

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read On a one-dimensional line, a two-candidate committee picked from rankings alone always contains the optimal single candidate for total distance, giving a 1-distortion of 1.

desk verdict Fresh bicriteria benchmark with clean line results, but the 2D lower bounds and the ordering lemma need repair before the paper is rigorous. read the letter →

arxiv 2412.10671 v1 pith:P3LR5YF7 submitted 2024-12-14 cs.GT cs.DS

Kiarash Banihashem , Diptarka Chakraborty , Shayan Chashm Jahan , Iman Gholami , MohammadTaghi Hajiaghayi , Mohammad Mahdavi , Max Springer This is my paper · ORCID
classification cs.GTcs.DS MSC 91B1291B1468W25
keywords metricdistortionmultiwinnervotingordinalpreferenceslinecommitteeelectionruleapproximationratiosocialchoice
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies how much efficiency is lost when a voting rule sees only voters' rankings of candidates, not their true distances. It asks whether electing a small committee, and charging each voter only the distance to the nearest elected candidate, can beat the distortion-3 barrier that is unavoidable when exactly one winner is chosen. The main answer is yes on a line metric: with just two elected candidates, an algorithm can guarantee that the committee contains the single candidate minimizing total distance, giving a ratio of 1 to the optimal single candidate. The same ordering technique gives ratios 1, 3/2, and 2 for the maximum-distance objective with committees of sizes four, three, and two, all optimal. For general metrics, the paper shows m-1 candidates achieve total-distance ratio 1+2/(m-1), matching a lower bound, and that no deterministic rule can beat ratio 3 for maximum distance even with m-1 candidates in the plane.

What carries the argument

The engine is the sorting algorithm SplitLine plus SortCandidates and SortVoters. Starting from a pivot voter's two nearest candidates, whose midpoint defines a pivot point, the algorithm repeatedly calls Determine: for any undetermined candidate c_k that some voter ranks above a determined candidate c_j, it places c_k on the opposite side of the pivot point if both pivot candidates are closer to that voter than c_j, and otherwise on the side of the nearer pivot candidate. The invariant that the determined set stays consecutive on the line makes each such inference sound, and the process ends with a core set C* that every voter prefers to every candidate outside it. Sorting C* by the pivot voter's order and ordering voters lexicographically by their preferences over C* yields the voter order used to locate the median; the load-bearing inequality is $\mathrm{cost}_s(c_2) \le \mathrm{cost}_s(c_3)+(|V_3|-|V_1|)\,d(c_2,c_3)$, whose mirror for $c_1$ decides which of the two outer candidates can be discarded.

What would settle it

Enumerate all small line instances, say five candidates and four voters with distinct positions, and run the public algorithm; if any instance yields a determined set that is not consecutive on the true line, or a candidate placed on the wrong side of the pivot midpoint while every voter's ranking matches the distances, Lemma 13 fails and the 2-committee guarantee collapses. A cheaper check is to search for a voter v_i, a determined c_j, and an undetermined c_k with c_k ranked above c_j while both pivot candidates are closer to v_i than c_j, yet c_k and c_j lie on the same side of the pivot point in the true coordinates.

Watch

Extended reading notes

Core claim

The central claim is Theorem 20: there is a deterministic voting rule for the 2-committee sum-cost election on the line metric that always outputs a pair of candidates containing an optimal single candidate, so its 1-distortion is 1. The proof first recovers, from ordinal preferences alone, the left-to-right order of a core set of candidates and the order of voters with respect to that set; then it finds the median voter, takes that voter's closest candidate plus its immediate left and right neighbours, and shows one of these three is optimal by the median-voter property for sum costs. A counting argument on which side has more voters lets the rule drop one neighbour and still keep the optimal candidate. The paper also establishes matching upper and lower bounds for the max-cost objective on the line and for general and 2-D Euclidean metrics.

Load-bearing premise

The ordering algorithm assumes that the voters' rankings really come from distances on a line, and that the geometric inference in Lemma 13, which decides an undetermined candidate's side of the pivot from whether both pivot candidates are closer to some voter, is valid whenever the determined set is consecutive.

Editorial extensions

If this is right

  • A two-candidate election on a line is never worse than the best single winner under total distance, even though each voter's distance is measured to the closer of the two.
  • Three candidates give a 3/2-approximation and four give the exact optimum under maximum distance, with matching lower bounds showing fewer candidates cannot do better.
  • The lower bounds imply that on the line the committee size, not the amount of preference detail, is what buys accuracy: in the plane, no number of candidates below m improves the max-cost ratio beyond 3.
  • The core-ordering routine gives a general way to recover geometric order from ordinal data, so it can be reused for other line-metric voting problems.
  • The bicriteria framework establishes a new benchmark for multiwinner voting: comparing a k-committee to the best single winner, rather than to the best k-committee.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension left implicit by the paper is to tree metrics or metrics of bounded doubling dimension; the consecutive-interval structure used in the line proof fails there, so a different invariant would be needed.
  • The two-candidate shortlist could serve as a preprocessing step when distances are privately known: any mechanism that then chooses between the two candidates inherits the guarantee that one of them is optimal.
  • The recursive lower-bound construction suggests that on a line, choosing sub-exponentially many candidates may still force unbounded distortion against an optimal committee, which would connect to the literature on small winning sets.
  • A testable empirical extension is to run the ranking-only two-candidate rule on synthetic line elections with known positions and compare its cost with k-median on the true positions; the gap should be zero whenever the optimal candidate lies beside the median voter.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces a bicriteria perspective on metric distortion: a voting rule selects a committee of k candidates and is evaluated by comparing the resulting social cost (sum or max over voters of distance to the nearest selected candidate) against the cost of the best single candidate. The main positive claim is that on the line metric, a 2-committee with the sum objective achieves 1-distortion 1, i.e., it always contains an optimal candidate; for the egalitarian objective, committees of size 4, 3, and 2 achieve distortions 1, 1.5, and 2, respectively, all tight. For general metrics, an (m-1)-committee achieves sum-cost distortion 1+2/(m-1), with a matching 2D Euclidean lower bound. The upper-bound arguments rely on an algorithm that recovers the order of a 'core' subset of candidates and the order of voters from ordinal preferences on a line.

Significance. If established, the results give the first bicriteria treatment of metric distortion and show that allowing a constant-sized committee can bypass the classical factor-3 barrier for single-winner distortion on the line. The proposed ordering-recovery algorithm is a technically interesting tool that could be useful beyond this paper. The paper also contains several concrete lower-bound constructions. However, the manuscript in its current form has load-bearing gaps in both the ordering-recovery proof and the Euclidean lower-bound constructions, so the advertised theorems are not yet established as written.

major comments (4)
  1. [§3.1, Lemma 13, Case 2] The proof of Case 2 uses an invalid 'without loss of generality' assumption: it assumes c1 precedes c2 in voter vi's ordinal preference, but c1 and c2 are fixed as the left and right pivot candidates, so if vi lies to the right of the pivot point, vi legitimately prefers c2 to c1. The missing case (c2≻i c1) is not analyzed, and it is exactly the configuration where Algorithm 2's 'else' branch would output R. Because the Determine inference is the engine behind the recovered ordering (Theorem 10), Lemma 19, and hence the central 2-committee result (Theorem 20), this gap is load-bearing and must be repaired.
  2. [§3, Algorithm 3, line 8] The pseudocode calls Determine(≻i, ck, cj, L, R, c1, c2), but the function signature in Algorithm 2 is Determine(≻i, cj, ck, L, R, c1, c2) with the convention that the second argument is the determined candidate and the third is the undetermined one, and ck≻i cj. As written, the arguments are swapped, so the procedure would treat the already-determined cj as the candidate to place and would not implement the intended inference. The correctness claim for SplitLine is therefore not backed by the provided code.
  3. [§5.1, Theorem 31 (and §5.2, Theorem 32)] The lower-bound family I0 uses candidates ci at (−ℓ, (2i−1)/(2m)) and voters vi at (0, (2i−1)/(2m)). With this uniform y-spacing, voter vi (for i≥2) is at distance sqrt(ℓ² + ((i−1)/m)²) from c1 and at distance sqrt(ℓ² + (1/m)²) from c_{i+1}; for i>2 the latter is strictly smaller, so c_{i+1} is preferred to c1, contradicting the claimed profile ci≻ci−1≻...≻c1≻ci+1≻...≻cm. Thus the instances do not realize the ordinal preferences that the indistinguishability argument requires. The same issue affects Theorem 32. The construction may be repairable with exponentially spaced y-coordinates, but as written the lower bounds are not valid.
  4. [§4.2, Lemmas 23–24 and Theorem 26] The symbols vl and vr are inconsistently defined as 'leftmost and rightmost candidates' in Lemma 23 and as 'leftmost and rightmost voters' in Theorem 26 and Figure 5; Lemma 24 then refers to 'candidates placed between vl and vr' when vl and vr are voters. This notational ambiguity makes the proofs of Lemma 24 and Theorem 26 hard to follow and should be clarified with distinct names for the extreme voters and the extreme candidates.
minor comments (5)
  1. [§3, Algorithm 3, line 7] The condition 'c_j∈ Det' uses an undefined symbol Det; it should be C* (the current set of determined candidates).
  2. [§3, Algorithm 1] The function name 'SortCandiatesAndVoters' is misspelled; also the variable name in the comment 'SC' is not defined consistently.
  3. [§4.1, Lemma 19] The proof says 'we consider the median voter v' and then uses 'v's closest candidate c' without explicitly handling ties among multiple median voters or multiple equidistant closest candidates; the argument goes through if ties are broken consistently, but this should be stated.
  4. [§5.1, Theorem 31 proof] The sentence 'the distance to any other candidate ci remains the same as in I0' for the moved voter vj is false: moving vj from x=0 to x=ℓ changes the x-separation to every candidate ci (i≠j) from ℓ to 2ℓ. The relative order among those candidates is preserved, but the stated reason is incorrect.
  5. [General] Several typos and minor formatting issues appear throughout (e.g., 'offer', 'prefernece', inconsistent use of 'ci' vs 'ck' in Algorithm 2 comments); a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the upper-bound algorithm is derived from ordinal preferences without fitted parameters, and the lower bounds are adversarial constructions.

full rationale

The paper's central derivation is self-contained. Section 3 (Theorem 10, Lemmas 11-17) proves that an iterative algorithm reconstructs a core subset of candidates and their order on the line from ordinal preferences alone. The 'core' property (Definition 5) is a target to be found, not an assumption: the algorithm uses the preference profile to expand C* and then proves that expansion yields a core. The selection results (Lemma 19, Theorem 20, and the max-cost Theorems 26-28) then use the recovered order plus elementary median/triangle-inequality arguments on the line metric; they do not fit any parameter to the quantity being predicted. The comparison to an optimal single candidate is external: voters' underlying distances are never revealed to the algorithm, so proving that a two-candidate shortlist contains an optimal candidate is substantive, not definitional. The lower bounds (Theorems 29, 31-34) are explicit instance families that any deterministic rule must fail on, with the same ordinal profile across geometrically different embeddings; these do not reduce to the algorithm's own outputs. There are no load-bearing self-citations: the references to prior metric-distortion results are external and standard. The reader's and skeptic's concerns about Lemma 13 are about proof correctness or a possible gap in the WLOG argument, not about circularity: a false or missing step in a proof is not the same as the result being equivalent to its inputs by construction. Accordingly, no circular step meeting the quoted-reduction standard was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented physical or conceptual entities. The main load-bearing ingredients are the metric-consistency model, the line-metric assumption, standard geometric facts about intervals on a line, the median-voter lemma, and the realizability of the 2D lower-bound constructions. The last of these is violated by the paper's own coordinates, which is the principal soundness concern.

assumptions (5)
  • domain assumption The true metric d is consistent with the ordinal preferences: if c_j is preferred to c_k by voter v_i, then d(v_i,c_j) <= d(v_i,c_k).
    This is the standard metric distortion model, stated in Section 2 under the definition of consistency.
  • domain assumption For the main upper bounds, voters and candidates lie on a real line.
    The line-metric assumption is explicit in Section 2 and is necessary for the median-voter arguments and the sorting algorithm.
  • standard math For any voter on a line, the k most preferred candidates form a consecutive subsequence of candidates.
    Lemma 11, used throughout Section 3 to justify the SplitLine determination rules and the consecutiveness of the determined set.
  • standard math The median voter theorem for sum of distances on a line: an optimal single candidate is one of the candidates adjacent to a median voter.
    Lemma 18, used in the 3-committee and 2-committee sum-cost upper bounds to identify the candidate set containing an optimum.
  • ad hoc to paper The 2D lower-bound instances with y-coordinates y_i = (2i-1)/(2m) realize the stated ordinal preferences.
    This is asserted in Theorems 31 and 32 but is false for m >= 4: under the given coordinates, d(v_i,c_{i-2}) > d(v_i,c_{i+1}) while the asserted preference order has c_{i-2} above c_{i+1}. The lower-bound proofs depend on this realizability.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bi-Criteria Metric Distortion." pith.science (2026). https://pith.science/paper/P3LR5YF7

@misc{pith2026241210671,
  author       = {Pith},
  title        = {Pith review of: Bi-Criteria Metric Distortion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3LR5YF7}},
  note         = {Machine review of arXiv:2412.10671}
}
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 approximation compared to an optimal candidate by relying on a committee of $k$ candidates ($k \ge 1$), where the cost of a voter is defined as its distance to the closest candidate in the committee. We answer this affirmatively in the case of line metrics, demonstrating that with $O(1)$ candidates, it is possible to achieve optimal cost. Our results extend to both utilitarian and egalitarian objectives, providing new upper bounds for the problem. We complement our results with lower bounds for both the line and 2-D Euclidean metrics.

Figures

Figures reproduced from arXiv: 2412.10671 by the authors.

Figure 1
Figure 1. This figure is the illustration of the succession of [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. For a voter vi , if we have c1 ≻i ck, c2 ≻i ck, and ck ≻i ck, then ck and cj cannot both be in the same side of p. The figure above illustrates this contradiction, while the one below shows that they can be on opposite sides. Case 2: At least one of c1 and c2 is positioned after ck in the ordinal preference of vi . Without loss of generality, assume that c1 precedes c2 in the ordinal preference of vi . Consider the … view at source ↗
Figure 3
Figure 3. For a voter vi , if we have c1 ≻i c2 and ck ≻i c2, then ck is in L. The figure above illustrates that if ck were in R, then c2 would be in the consecutive subsequence of c1 and ck which is a contradiction. On the other hand, the one below shows that ck must be in L. Lemma 14. In Algorithm 3, in the beginning of each iteration (Line 5), all determined candidates, denoted as C ∗ , form a consecutive subsequence of all… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: A figure illustrating three candidates c1, c2, and c3 along with the possible locations of voters closest to each candidate, V1, V2, and V3. Voters v1, v2 and v3 show examples of voters in each set. prefer the corresponding candidate to the other two, as illustrated in…
Figure 5
Figure 5. Figure 5: Illustration of the candidates cl and cr , and the voters vl and vr , demonstrating why choosing these candidates does not achieve a distortion of 1 Proof. Since v is a voter between vl and vr (recall vl is the left-most and vr is the right-most voter), we have: d(vl ,…
Figure 6
Figure 6. Figure 6: Illustration of the candidates cl and cr positioned between the voters vl and vr . In this case, we achieve the optimal answer instead of a two approximation. as cl lies between vl and copt and cr lies between vr and copt. By Lemma 25, we know that for every voter v, e…
Figure 7
Figure 7. Figure 7: Figures of the lower bound instances. In (a), all ca [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Figures of the lower bound instances for the max obj [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Possible locations of the voters and candidates in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Possible locations of the voters and candidates i [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 28 canonical work pages

  1. [1]

    Alamdari and D

    S. Alamdari and D. Shmoys. A bicriteria approximation al gorithm for the k-center and k-median prob- lems. In Approximation and Online Algorithms: 15th International W orkshop, WAOA 2017, Vienna, Austria, September 7–8, 2017, Revised Selected Papers 15 , pages 66–75. Springer, 2018

    work page 2017

  2. [2]

    Anagnostides, D

    I. Anagnostides, D. Fotakis, and P . Patsilinakos. Dimen sionality and coordination in voting: The distortion of stv. In Proceedings of the AAAI Conference on Artificial Intelligen ce, volume 36, pages 4776–4784, 2022

    work page 2022

  3. [3]

    Anshelevich, O

    E. Anshelevich, O. Bhardwaj, E. Elkind, J. Postl, and P . S kowron. Approximating optimal social choice under metric preferences. Artificial Intelligence , 264:27–51, 2018

    work page 2018

  4. [4]

    Anshelevich, A

    E. Anshelevich, A. Filos-Ratsikas, C. Jerrett, and A. A. V oudouris. Improved metric distortion via threshold approvals. In Proceedings of the AAAI Conference on Artificial Intelligen ce, volume 38, pages 9460–9468, 2024

    work page 2024

  5. [5]

    D. Black. On the rationale of group decision-making. Journal of political economy , 56(1):23–34, 1948

    work page 1948

  6. [6]

    Caragiannis, E

    I. Caragiannis, E. Micha, and J. Peters. Can a few decide f or many? the metric distortion of sortition. In F orty-first International Conference on Machine Learning

    work page

  7. [7]

    Caragiannis, N

    I. Caragiannis, N. Shah, and A. A. V oudouris. The metric d istortion of multiwinner voting. Artificial Intelligence, 313:103802, 2022

    work page 2022

  8. [8]

    Charikar, A

    M. Charikar, A. Lassota, P . Ramakrishnan, A. V etta, and K . Wang. Six candidates su ffice to win a voter majority. arXiv preprint arXiv:2411.03390, 2024

    arXiv 2024

Show all 31 references
  1. [9]

    Charikar and P

    M. Charikar and P . Ramakrishnan. Metric distortion boun ds for randomized social choice. In Proceed- ings of the 2022 Annual ACM-SIAM Symposium on Discrete Algor ithms (SODA), pages 2986–3004. SIAM, 2022

    2022

  2. [10]

    Charikar, P

    M. Charikar, P . Ramakrishnan, K. Wang, and H. Wu. Breaki ng the metric voting distortion barrier. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discre te Algorithms (SODA) , pages 1621–1640. SIAM, 2024. 26

    2024

  3. [11]

    X. Chen, M. Li, and C. Wang. Favorite-candidate voting f or eliminating the least popular candidate in a metric space. In Proceedings of the AAAI Conference on Artificial Intelligen ce, volume 34, pages 1894–1901, 2020

    1901

  4. [12]

    Elkind and P

    E. Elkind and P . Faliszewski. Recognizing 1-euclidean preferences: An alternative approach. In International Symposium on Algorithmic Game Theory , pages 146–157. Springer, 2014

    2014

  5. [13]

    Elkind, P

    E. Elkind, P . Faliszewski, P . Skowron, and A. Slinko. Pr operties of multiwinner voting rules. Social Choice and Welfare, 48:599–632, 2017

    2017

  6. [14]

    Elkind, J

    E. Elkind, J. Lang, and A. Sa ffidine. Condorcet winning sets. Social Choice and Welfare, 44(3):493– 517, 2015

    2015

  7. [15]

    J. M. Enelow and M. J. Hinich. The spatial theory of voting: An introduction . CUP Archive, 1984

    1984

  8. [16]

    Esco ffier, J

    B. Esco ffier, J. Lang, and M. Öztürk. Single-peaked consistency and it s complexity. In ECAI 2008, pages 366–370. IOS Press, 2008

    2008

  9. [17]

    Faliszewski, P

    P . Faliszewski, P . Skowron, A. Slinko, and N. Talmon. Multiwinner voting: A new challenge for social choice theory. Trends in computational social choice, 74(2017):27–47, 2017

    2017

  10. [18]

    Feldman, A

    D. Feldman, A. Fiat, M. Sharir, and D. Segev. Bi-criteri a linear-time approximations for generalized k-mean/median/center. In Proceedings of the twenty-third annual symposium on Comput ational geom- etry, pages 19–26, 2007

    2007

  11. [19]

    Feldman, A

    M. Feldman, A. Fiat, and I. Golomb. On voting and facilit y location. In Proceedings of the 2016 ACM Conference on Economics and Computation , pages 269–286, 2016

    2016

  12. [20]

    Gkatzelis, D

    V . Gkatzelis, D. Halpern, and N. Shah. Resolving the opt imal metric distortion conjecture. In 2020 IEEE 61st Annual Symposium on F oundations of Computer Science (FOCS), pages 1427–1438. IEEE, 2020

    2020

  13. [21]

    A. Goel, R. Hulett, and A. K. Krishnaswamy. Relating met ric distortion and fairness of social choice rules. In Proceedings of the 13th Workshop on Economics of Networks, S ystems and Computation , pages 1–1, 2018

    2018

  14. [22]

    A. Goel, A. K. Krishnaswamy, and K. Munagala. Metric dis tortion of social choice rules: Lower bounds and fairness properties. In Proceedings of the 2017 ACM Conference on Economics and Com- putation, pages 287–304, 2017

    2017

  15. [23]

    D. Kempe. An analysis framework for metric voting based on lp duality. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 34, pages 2079–2086, 2020

    2020

  16. [24]

    F. E. Kizilkaya and D. Kempe. Generalized veto core and a practical voting rule with optimal metric distortion. In Proceedings of the 24th ACM Conference on Economics and Comp utation, pages 913– 936, 2023

    2023

  17. [25]

    F. E. Kizilkaya and D. Kempe. Plurality veto: A simple vo ting rule achieving optimal metric distortion, 2023

    2023

  18. [26]

    Lassota, A

    A. Lassota, A. V etta, and B. von Stengel. The condorcet d imension of metric spaces. arXiv preprint arXiv:2410.09201, 2024

    2024 arXiv

  19. [27]

    Mahdian, Y

    M. Mahdian, Y . Y e, and J. Zhang. Approximation algorith ms for metric facility location problems. SIAM Journal on Computing, 36(2):411–432, 2006. 27

    2006

  20. [28]

    J. A. Mirrlees. An exploration in the theory of optimum i ncome taxation. The review of economic studies, 38(2):175–208, 1971

    1971

  21. [29]

    A. D. Procaccia and J. S. Rosenschein. The distortion of cardinal preferences in voting. In International Workshop on Cooperative Information Agents, pages 317–331. Springer, 2006

    2006

  22. [30]

    Pulyassary and C

    H. Pulyassary and C. Swamy. On the randomized metric dis tortion conjecture. arXiv preprint arXiv:2111.08698, 2021

    2021 arXiv

  23. [31]

    D. Wei. A constant-factor bi-criteria approximation g uarantee for k-means ++. Advances in neural information processing systems, 29, 2016. 28

    2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.