Abstract
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 L-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 L-p metrics, and more particularly on Manhattan (L-1), Euclidean (L-2), Hamming (L-0) and Chebyshev (L-infinity) metrics of dimension log n and above. 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 L-infinity 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 L-1 metric and to a factor of 1.17 in the L-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 L-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 L-p metrics. Our hardness result given in (1) above, is under the standard NP is not equal to 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 L-1 and L-infinity 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.