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Unformatted text preview: Inapproximability for planar embedding problems Jeff Edmonds 1 Anastasios Sidiropoulos 2 Anastasios Zouzias 3 1 York University 2 TTIChicago 3 University of Toronto January, 2010 A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 1 / 21 Metric Space M = ( X , D ) is a metric space. X is a set. D is a distance function on X , i.e., satisfies triangle inequality. Examples Any normed space. Graphs with shortest path distance. ... ... ... A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 2 / 21 Embedding between metric spaces Given M = ( X , d X ) and M ′ = ( Y , d Y ) . Embedding f : X mapsto→ Y . Metric distortion f has distortion α if ∀ x 1 , x 2 ∈ X d X ( x 1 , x 2 ) ≤ d Y ( f ( x 1 ) , f ( x 2 )) ≤ α · d X ( x 1 , x 2 ) . dist ( f ) = maxexp( f ) × maxcontr( f ) Wellstudied subject: Worst case distortion. Relative Embeddings: Given X , find nearoptimal embedding f : X mapsto→ Y efficiently. A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 3 / 21 Embedding between metric spaces Given M = ( X , d X ) and M ′ = ( Y , d Y ) . Embedding f : X mapsto→ Y . Metric distortion f has distortion α if ∀ x 1 , x 2 ∈ X d X ( x 1 , x 2 ) ≤ d Y ( f ( x 1 ) , f ( x 2 )) ≤ α · d X ( x 1 , x 2 ) . dist ( f ) = maxexp( f ) × maxcontr( f ) Wellstudied subject: Worst case distortion. Relative Embeddings: Given X , find nearoptimal embedding f : X mapsto→ Y efficiently. A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 3 / 21 Computational problems (Approximate min. distortion) Bijection and Injection Bijection Given two finite metric spaces X , Y of the same size n . Approximate the minimum distortion bijection f : X mapsto→ Y . Introduced in [KRS04]. Injection Given finite metric space X of size n and infinite metric space Y with fixed dimensionality. Approximate the minimum distortion injection of f : X mapsto→ Y . Remark: Although different problems, share the same approximability. Notation α vs. ̙ : Given X it is NPhard to check if ∃ f : X mapsto→ Y with distortion ≤ α or every f has distortion > ̙ . Notice that 1 ≤ α < ̙ . A. Zouzias (University of Toronto) Inapproximability for planar embedding problems SODA 2010 4 / 21 Computational problems (Approximate min. distortion) Bijection and Injection Bijection Given two finite metric spaces X , Y of the same size n . Approximate the minimum distortion bijection f : X mapsto→ Y . Introduced in [KRS04]. Injection Given finite metric space X of size n and infinite metric space Y with fixed dimensionality. Approximate the minimum distortion injection of f : X mapsto→ Y ....
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 Winter '12
 Edmonds
 Distance, Computational complexity theory, Metric space, Topological space, planar embedding problems

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