2026-07-22T04:52:49Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8734 2024-03-06T10:42:31Z ddc:004 open_access

Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces Carpenter, Timothy Fomin, Fedor V. Lokshtanov, Daniel Saurabh, Saket Sidiropoulos, Anastasios Metric embeddings minimum-distortion embeddings 1-dimensional simplicial complex Fixed-parameter tractable algorithms Approximation algorithms We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H=K_2, and embedding into the cycle is the case H=K_3, where K_k denotes the complete graph on k vertices. For this problem we give - an approximation algorithm, which in time f(H)* poly (n), for some function f, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c); - an exact algorithm, which in time f'(H, c)* poly (n), for some function f', either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Timothy Carpenter and Fedor V. Fomin and Daniel Lokshtanov and Saket Saurabh and Anastasios Sidiropoulos 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.21 urn:nbn:de:0030-drops-87344 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.21 eng https://creativecommons.org/licenses/by/3.0/legalcode