Inapproximability for planar embedding problems

We consider the problem of computing a minimumdistortion bijection between two point-sets in R2 . We prove the first non-trivial inapproximability result for this problem, for the case when the distortion is constant. More precisely, we show that there exist constants 0 < < , such that it is NP-hard to distinguish between spaces for which the distortion is either at most , or at least , under the Euclidean norm. This addresses a question of Kenyon, Rabani and Sinclair [KRS04], and extends a result due to Papadimitriou and Safra [PS05], who gave inapproximability for point-sets in R3 . We also apply similar ideas to the problem of computing a minimum-distortion embedding of a finite metric space into R2 . We obtain an analogous inapproximability result under the norm for this problem. Inapproximability for the case of constant distortion was previously known only for dimension at least 3 [MS08].

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: Jeff Edmonds, Anastasios Sidiropoulos, Anastasios Zouzias
• Keywords: Constant Distortion, Discrete Algorithms, Euclidean Norm, Finite Metric Space, SODA 2010