INAPPROXIMABILITY FOR METRIC EMBEDDINGS INTO R

We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into Rd, where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n1/12 is NP-hard. From this result we derive inapproximability with a factor roughly n1/(22d−10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Väisälä). For d ≥ 3, we obtain a stronger inapproximability result by a different reduction: assuming P =NP, no polynomial-time algorithm can distinguish between spaces embeddable in Rd with constant distortion from spaces requiring distortion at least nc/d, for a constant c > 0. The exponent c/d has the correct order of magnitude, since every n-point metric space can be embedded in Rd with distortion O(n2/d log n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in R2, while all not too large subspaces of it embed almost isometrically.

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