Cuts, Trees and ℓ1-Embeddings of Graphs*

Motivated by many recent algorithmic applications, the paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred where the graph is embedded into l/sub 1/ space. The main results are: 1. Explicit constant-distortion embeddings of all series parallel graphs, and all graphs with bounded Euler number. These are thus the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, we obtain algorithms to 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 l/sub 1/-metrics known as "dominating tree metrics". We also show a lower bound of /spl Omega/(log n) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low tree width, and excludes the possibility of using them to explore the finer structure of l/sub 1/-embeddability.

[1]  Yossi Azar,et al.  Buy-at-bulk network design , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[2]  Alexander Schrijver,et al.  Homotopic routing methods , 1990 .

[3]  Moses Charikar,et al.  The finite capacity dial-a-ride problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[4]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[5]  R. Ravi,et al.  On approximating planar metrics by tree metrics , 2001, Inf. Process. Lett..

[6]  Philip N. Klein,et al.  Excluded minors, network decomposition, and multicommodity flow , 1993, STOC.

[7]  Satish Rao,et al.  An approximate max-flow min-cut relation for undirected multicommodity flow, with applications , 1995, Comb..

[8]  J. Bourgain.  The metrical interpretation of superreflexivity in banach spaces , 1986 .

[9]  Haruko Okamura,et al.  Multicommodity flows in planar graphs , 1981, J. Comb. Theory, Ser. B.

[10]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[11]  David B. Shmoys,et al.  Cut problems and their application to divide-and-conquer , 1996 .

[12]  Chuan Yi Tang,et al.  A polynomial time approximation scheme for minimum routing cost spanning trees , 1998, SODA '98.

[13]  Yuval Rabani,et al.  An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm , 1998, SIAM J. Comput..

[14]  Makoto Imase,et al.  Dynamic Steiner Tree Problem , 1991, SIAM J. Discret. Math..

[15]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[16]  Uriel Feige,et al.  Approximating the Bandwidth via Volume Respecting Embeddings , 2000, J. Comput. Syst. Sci..

[17]  Satish Rao,et al.  Small distortion and volume preserving embeddings for planar and Euclidean metrics , 1999, SCG '99.

[18]  Andrew Tomkins,et al.  A polylog(n)-competitive algorithm for metrical task systems , 1997, STOC '97.

[19]  Ran Raz,et al.  Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs , 1998, Discret. Comput. Geom..

[20]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[21]  J. Matousek,et al.  On embedding trees into uniformly convex Banach spaces , 1999 .

[22]  V. Chepoi,et al.  A Note on Circular Decomposable Metrics , 1998 .

[23]  Santosh S. Vempala,et al.  Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems , 2000, Theor. Comput. Sci..

[24]  Noga Alon,et al.  A Graph-Theoretic Game and Its Application to the k-Server Problem , 1995, SIAM J. Comput..

[25]  Sudipto Guha,et al.  Approximating a finite metric by a small number of tree metrics , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[26]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[27]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[28]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[29]  Sudipto Guha,et al.  Rounding via Trees : Deterministic Approximation Algorithms forGroup , 1998 .

[30]  Cynthia A. Phillips,et al.  Finding minimum-quotient cuts in planar graphs , 1993, STOC.

[31]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[32]  Frank Thomson Leighton,et al.  A Framework for Solving VLSI Graph Layout Problems , 1983, J. Comput. Syst. Sci..