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Distortion is Fixed Parameter Tractable

Published:01 November 2013Publication History
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Abstract

We study low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Let M = M(G) be the shortest path metric of an edge-weighted graph G, with the vertex set V(G) and the edge set E(G), on n vertices. We give the first fixed parameter tractable algorithm that for an unweighted graph metric M and integer d either constructs an embedding of M into the line with distortion at most d, or concludes that no such embedding exists. Our algorithm requires O(nd4(2d + 1)2d) time which is a significant improvement over the best previous algorithm that runs in time O(n4d+2dO(1)). Because of its apparent similarity to the notoriously hard Bandwidth Minimization problem, we find it surprising that this problem turns out to be fixed parameter tractable.

We extend our results on embedding unweighted graph metric into the line in two ways. First, we give an algorithm to construct small-distortion embeddings of weighted graph metrics. The running time of our algorithm is O(n(dW)4(2d + 1)2dW), where W is the largest edge weight of the input graph. To complement this result, we show that the exponential dependence on the maximum edge weight is unavoidable. In particular, we show that deciding whether a weighted graph metric M(G) with maximum weight W < |V(G)| can be embedded into the line with distortion at most d is NP-complete for every fixed rational d ≥ 2. This rules out any possibility of an algorithm with running time O((nW)h(d)) where h is a function of d alone. Second, we consider more general host metrics for which analogous results hold. In particular, we prove that for any tree T with maximum degree Δ, embedding M into a shortest path metric of T is fixed parameter tractable, parameterized by (Δ, d).

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 5, Issue 4
      November 2013
      103 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2539126
      Issue’s Table of Contents

      Copyright © 2013 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 November 2013
      • Revised: 1 April 2013
      • Accepted: 1 April 2013
      • Received: 1 September 2012
      Published in toct Volume 5, Issue 4

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