TY - GEN

T1 - FPT algorithms for embedding into low complexity graphic metrics

AU - Ghosh, Arijit

AU - Kolay, Sudeshna

AU - Mishra, Gopinath

N1 - Publisher Copyright:
© Arijit Ghosh, Sudeshna Kolay, and Gopinath Mishra.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - The METRIC EMBEDDING problem takes as input two metric spaces (X,DX) and (Y,DY), and a positive integer d. The objective is to determine whether there is an embedding F: X → Y such that the distortion dF ≤ d. Such an embedding is called a distortion d embedding. In parameterized complexity, the METRIC EMBEDDING problem is known to be W-hard and therefore, not expected to have an FPT algorithm. In this paper, we consider the GEN-GRAPH METRIC EMBEDDING 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, DG) can be embedded, or bijectively embedded, into another unweighted graph metric (H,DH), 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 the connected treewidth αc 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.

AB - The METRIC EMBEDDING problem takes as input two metric spaces (X,DX) and (Y,DY), and a positive integer d. The objective is to determine whether there is an embedding F: X → Y such that the distortion dF ≤ d. Such an embedding is called a distortion d embedding. In parameterized complexity, the METRIC EMBEDDING problem is known to be W-hard and therefore, not expected to have an FPT algorithm. In this paper, we consider the GEN-GRAPH METRIC EMBEDDING 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, DG) can be embedded, or bijectively embedded, into another unweighted graph metric (H,DH), 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 the connected treewidth αc 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.

KW - Dynamic programming

KW - FPT

KW - Metric embedding

KW - Metric spaces

UR - http://www.scopus.com/inward/record.url?scp=85052533117&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2018.35

DO - 10.4230/LIPIcs.ESA.2018.35

M3 - Conference contribution

AN - SCOPUS:85052533117

SN - 9783959770811

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 26th European Symposium on Algorithms, ESA 2018

A2 - Bast, Hannah

A2 - Herman, Grzegorz

A2 - Azar, Yossi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 26th European Symposium on Algorithms, ESA 2018

Y2 - 20 August 2018 through 22 August 2018

ER -