NDLI: Inapproximability for Metric Embeddings into R^d

Content Provider	IEEE Xplore Digital Library
Author	Matousek, J. Sidiropoulos, A.
Copyright Year	2008
Description	Author affiliation: Dept. of Appl. Math., Charles Univ., Prague (Matousek, J.) \|\| Comput. Sci. & Artificial Intell. Lab., Massachusetts Inst. of Technol., Cambridge, MA (Sidiropoulos, A.)
Abstract	We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into R.d, where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly $n^{1/12}$ is NP-hard. From this result we derive inapproximability with factor roughly $n^{1/(22d-10)}$ for every fixed d ges 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 Vaisala). For d ges 3,we obtain a stronger inapproximability result by a different reduction: assuming PneNP, no polynomial- time algorithm can distinguish between spaces embeddable in R.d with constant distortion from spaces requiring distortion at least $n^{c/d},$ 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 $R^{d}$ with distortion $O(n^{2/d}$ $log^{3/2}$ 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 $R^{2},$ while all not too large subspaces of it embed almost isometrically.
Starting Page	405
Ending Page	413
File Size	337302
Page Count	9
File Format	PDF
ISBN	9780769534367
ISSN	02725428
DOI	10.1109/FOCS.2008.21
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-10-25
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Extraterrestrial measurements Computer science Polynomials Approximation algorithms Space technology Mathematics Artificial intelligence Laboratories USA Councils Embedded computing inapproximability embeddings hardness
Content Type	Text
Resource Type	Article

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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