An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Improved Approximation Algorithms for Minimum Weight Vertex Separators
SIAM Journal on Computing
Embeddings of Topological Graphs: Lossy Invariants, Linearization, and 2-Sums
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
On the geometry of graphs with a forbidden minor
Proceedings of the forty-first annual ACM symposium on Theory of computing
Coarse Differentiation and Multi-flows in Planar Graphs
Discrete & Computational Geometry
Bilipschitz snowflakes and metrics of negative type
Proceedings of the forty-second ACM symposium on Theory of computing
Flow-cut gaps for integer and fractional multiflows
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Multicommodity flows and cuts in polymatroidal networks
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
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The 2-sum embedding conjecture (Lee-Sidiropoulos, FOCS 2009) states that all the shortest-path metrics supported on a family of graphs F admit a uniformly bi-Lipschitz embedding into L1 if and only if the same property holds for the closure of F under edge sums. The problem has an equivalent formulation in terms of multi-commodity flow/cut gaps and appears to be a fundamental step in resolving the well-studied question of which families of graphs admit an approximate multi-commodity max-flow/min-cut theorem. The 2-sum conjecture is known to be true when F = {K3}, where Kn denotes the complete graph on n vertices; this is equivalent to the seminal result of Gupta, Newman, Rabinovich, and Sinclair (Combinatorica, 2004) on embeddings of series-parallel graphs into L1. For F = {K4}, the conjecture was confirmed by Chakrabarti, Jaffe, Lee, and Vincent (FOCS 2008). In the present paper, we prove the conjecture holds for any finite family of graphs F. In fact, we obtain a quantitatively optimal result: Any path metric on a graph formed by 2-sums of arbitrarily many copies of Kn embeds into L1 with distortion O(log n). This is a consequence of a more general theorem that characterizes the L1 distortion of the 2-sum closure of any family F in terms of constrained embeddings of the members of F.