Inapproximability of Clustering in Lp-metrics - Archive ouverte HAL Access content directly
Conference Papers Year : 2019

Inapproximability of Clustering in Lp-metrics

Vincent Cohen-Addad
Karthik Srikanta
  • Function : Author


Proving hardness of approximation for min-sum objectives is an infamous challenge. For classic problems such as the Traveling Salesman problem, the Steiner tree problem, or the k-means and k-median problems, the best known inapproximability bounds for p-metrics of dimension O(log n) remain well below 1.01. In this paper, we take a significant step to improve the hardness of approximation of the k-means problem in various p-metrics, and more particularly on 1 , 2 , Hamming and ∞ metrics of dimension Ω(log n). We show that it is hard to approximate the k-means objective in O(log n)-dimensional space: (1) To a factor of 3.94 in the ∞-metric when centers have to be chosen from a discrete set of locations (i.e., the discrete case). This improves upon the result of Guruswami and Indyk (SODA'03) who proved hardness of approximation for a factor less than 1.01. (2) To a factor of 1.56 in the 1-metric and to a factor of 1.17 in the 2-metric, both in the discrete case. This improves upon the result of Trevisan (SICOMP'00) who proved hardness of approximation for a factor less than 1.01 in both the metrics. (3) To a factor of 1.07 in the 2-metric, when centers can be placed at arbitrary locations, (i.e., the continuous case). This improves on a result of Lee-Schmidt-Wright (IPL'17) who proved hardness of approximation for a factor of 1.0013. We also obtain similar improvements over the state-of-the-art hardness of approximation results for the k-median objective in various p-metrics. Our hardness result given in (1) above, is under the standard NP = P assumption, whereas all the remaining results given above are under the Unique Games Conjecture (UGC). We can remove our reliance on UGC and prove standard NP-hardness for the above problems but for smaller approximation factors. Finally, we note that in order to obtain our result for the 1 and ∞-metrics in O(log n)-dimensional space we introduce an embedding technique which combines the transcripts of certain communication protocols with the geometric realization of certain graphs
Fichier principal
Vignette du fichier
main.pdf (388.82 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-02360762 , version 1 (13-11-2019)
hal-02360762 , version 2 (15-12-2019)


  • HAL Id : hal-02360762 , version 2


Vincent Cohen-Addad, Karthik Srikanta. Inapproximability of Clustering in Lp-metrics. FOCS'19 - 60th Annual IEEE Symposium on Foundations of Computer Science, Nov 2019, Baltimore, United States. ⟨hal-02360762v2⟩
179 View
637 Download


Gmail Facebook X LinkedIn More