Hardness of Embedding Metric Spaces of Equal Size

We study the problem embedding an n-point metric space into another n-point metric space while minimizing distortion. We show that there is no polynomial time algorithm to approximate the minimum distortion within a factor of i¾?((logn)1/4 i¾? i¾?) for any constant i¾?> 0, unless $\textnormal{NP} \subseteq \textnormal{DTIME}(n^{\textnormal{poly}(\log n))})$. We give a simple reduction from the METRIC LABELING problem which was shown to be inapproximable by Chuzhoy and Naor [10].

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