Inapproximability for planar embedding problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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We consider the problem of computing the smallestpossible distortion for embedding ofa given $n$-point metric space into $\R^d$, where $d$is \emph{fixed} (and small). For $d=1$, it was known thatapproximating the minimum distortion with a factor better than roughly $n^{1/12}$ is NP-hard. From this resultwe derive inapproximability withfactor roughly $n^{1/(22d-10)}$ for every fixed $d\ge 2$,by a conceptually very simple reduction. However,the proof of correctness involves a nontrivialresult in geometric topology (whose current proof isbased on ideas due to Jussi V\"ais\"al\"a).For $d\ge 3$, we obtain a stronger inapproximabilityresult by a different reduction: assuming P$\ne$NP,no polynomial-time algorithm can distinguish betweenspaces embeddable in $\R^d$ with constant distortionfrom spaces requiring distortion at least $n^{c/d}$,for a constant $c0$. The exponent $c/d$has the correct order of magnitude, sinceevery $n$-point metric space canbe embedded in $\R^d$ with distortion$O(n^{2/d}\log^{3/2}n)$ and such an embeddingcan be constructed in polynomial time byrandom projection.For $d=2$, we give an example of a metric space thatrequires a large distortion for embedding in $\R^2$,while all not too large subspaces of it embed almost isometrically.