Combinatorica

, Volume 24, Issue 2, pp 233–269

Cuts, Trees and ℓ1-Embeddings of Graphs*

  • Anupam Gupta†
  • Ilan Newman
  • Yuri Rabinovich
  • Alistair Sinclair‡
Original Paper

DOI: 10.1007/s00493-004-0015-x

Cite this article as:
Gupta†, A., Newman, I., Rabinovich, Y. et al. Combinatorica (2004) 24: 233. doi:10.1007/s00493-004-0015-x

Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ℓ1 space. The main results are:

1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, algorithms are obtained which approximate the sparsest cut in such graphs to within a constant factor.

2. A constant-distortion embedding of outerplanar graphs into the restricted class of ℓ1-metrics known as “dominating tree metrics”. A lower bound of Ω(log n) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics is also presented. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low treewidth, and excludes the possibility of using them to explore the finer structure of ℓ1-embeddability.

Mathematics Subject Classification (2000):

05C1205C8568R1090C27

Copyright information

© János Bolyai Mathematical Society 2004

Authors and Affiliations

  • Anupam Gupta†
    • 1
  • Ilan Newman
    • 2
  • Yuri Rabinovich
    • 3
  • Alistair Sinclair‡
    • 4
  1. 1.Department of Computer ScienceCarnegie Mellon UniversityPittsburghUSA
  2. 2.Computer Science DepartmentUniversity of HaifaHaifaIsrael
  3. 3.Computer Science DepartmentUniversity of HaifaHaifaIsrael
  4. 4.Computer Science Division, Soda HallUniversity of CaliforniaBerkeleyUSA