2 research outputs found
Metric embedding with outliers
We initiate the study of metric embeddings with \emph{outliers}. Given some
metric space we wish to find a small set of outlier points and either an isometric or a low-distortion embedding of
into some target metric space. This is a natural problem
that captures scenarios where a small fraction of points in the input
corresponds to noise.
For the case of isometric embeddings we derive polynomial-time approximation
algorithms for minimizing the number of outliers when the target space is an
ultrametric, a tree metric, or constant-dimensional Euclidean space. The
approximation factors are 3, 4 and 2, respectively. For the case of embedding
into an ultrametric or tree metric, we further improve the running time to
for an -point input metric space, which is optimal. We complement
these upper bounds by showing that outlier embedding into ultrametrics, trees,
and -dimensional Euclidean space for any are all NP-hard, as well
as NP-hard to approximate within a factor better than 2 assuming the Unique
Game Conjecture.
For the case of non-isometries we consider embeddings with small
distortion. We present polynomial-time \emph{bi-criteria}
approximation algorithms. Specifically, given some , let
denote the minimum number of outliers required to obtain an
embedding with distortion . For the case of embedding into
ultrametrics we obtain a polynomial-time algorithm which computes a set of at
most outliers and an embedding of the remaining points into an
ultrametric with distortion . For embedding a metric of
unit diameter into constant-dimensional Euclidean space we present a
polynomial-time algorithm which computes a set of at most
outliers and an embedding of the remaining points with distortion
Inapproximability for planar embedding problems
We consider the problem of computing a minimumdistortion bijection between two point-sets in R 2.Weprove 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 <α<β, such that it is NP-hard to distinguish between spaces for which the distortion is either at most α, oratleastβ, 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 R 3. We also apply similar ideas to the problem of computing a minimum-distortion embedding of a finite metric space into R 2. We obtain an analogous inapproximability result under the ℓ ∞ norm for this problem. Inapproximability for the case of constant distortion was previously known only for dimension at least 3 [MS08].