Abstract
Given metric spaces ( X , D X ) and ( Y , D Y ), an embedding F : X → Y is an injective mapping from X to Y . Expansion e F and contraction c F of an embedding F : X → Y are defined as e F = max x ; 1 , x 2 (≠ x 1 ) ∈ X D Y ( F ( x 1 ), F ( x 2 ))/ D X ( x 1 , x 2 ) and c F = max x 1 , x 2 (≠ x 1 ) ∈ X D X ( x 1 , x 2 )/ D Y ( F ( x 1 ), F ( x 2 )), respectively, and distortion d F is defined as d F = e F ⋅ c F . Observe that d F ≥ 1. An embedding F : X → Y is noncontracting if c F ≤ 1. When d =1, then F is isometry . The M etric E mbedding problem takes as input two metric spaces ( X , D X ) and ( Y , D Y ), and a positive integer d . The objective is to determine whether there is an embedding F : X → Y such that d F ≤ d . Such an embedding is called a distortion d embedding . The bijective M etric E mbedding problem is a special case of the M etric E mbedding problem where ∣ X ∣ = ∣ Y ∣. In parameterized complexity, the M etric E mbedding problem, in full generality, is known to be W-hard and, therefore, not expected to have an FPT algorithm. In this article, we consider the G en -G raph M etric E mbedding problem, where the two metric spaces are graph metrics. We explore the extent of tractability of the problem in the parameterized complexity setting. We determine whether an unweighted graph metric ( G , D G ) can be embedded, or bijectively embedded, into another unweighted graph metric ( H , D H ), where the graph H has low structural complexity. For example, H is a cycle, or H has bounded treewidth or bounded connected treewidth. The parameters for the algorithms are chosen from the upper bound d on distortion, bound Δ on the maximum degree of H , treewidth α of H , and connected treewidth α c of H . Our general approach to these problems can be summarized as trying to understand the behavior of the shortest paths in G under a low-distortion embedding into H , and the structural relation the mapping of these paths has to shortest paths in H .
Highlights
Given metric spaces (X, DX ) and (Y, DY ), an embedding F : X → Y is an injective mapping from X to Y
The problem of low distortion embedding of a metric space into a simple metric space has been extensively studied in Mathematics and Computer Science
We further investigate the problem of embedding a general graph metric (G, DG) into a low complexity graph metric (H, DH ) with distortion at most d
Summary
Given metric spaces (X, DX ) and (Y, DY ), an embedding F : X → Y is an injective mapping from X to Y. Fellows et al [7] extended this result to give an algorithm for the problem of finding a non-contracting embedding of unweighted graphs into bounded degree trees with distortion at most d in O(n2 · |V (T )|) · 2O((5d)∆d+1 ·d) time, where V (T ) denotes the vertex set of the tree and where the maximum degree in T is bounded by ∆. These graphs are more general than cycles and have constant treewidth, but they do not have bounded connected treewidth Even for this very structured graph class, by virtue of the graphs having long geodesic cycles, we needed to develop completely new ideas in order to find low distortion embeddings into generalized theta graphs via FPT algorithms. For full details please refer to the full version of the paper [8]
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