Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Approximation algorithms for Steiner and directed multicuts
Journal of Algorithms
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Preemptive Scheduling with Release Times, Deadlines, and Due Times
Journal of the ACM (JACM)
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Divide-and-conquer approximation algorithms via spreading metrics
Journal of the ACM (JACM)
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications
SIAM Journal on Computing
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximating Directed Multicuts
Combinatorica
Improved approximation for directed cut problems
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On Average Distortion of Embedding Metrics into the Line
Discrete & Computational Geometry
Combinatorial algorithms for wireless information flow
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
Polynomial flow-cut gaps and hardness of directed cut problems
Journal of the ACM (JACM)
Multicommodity flows and cuts in polymatroidal networks
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Wireless Network Information Flow: A Deterministic Approach
IEEE Transactions on Information Theory
A Max-Flow/Min-Cut Algorithm for Linear Deterministic Relay Networks
IEEE Transactions on Information Theory
Multicommodity flows and cuts in polymatroidal networks
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
A node-capacitated okamura-seymour theorem
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
On the 2-sum embedding conjecture
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [20] and Hassin [15] in the single-commodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the well-known maxflow-mincut theorem holds in this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish poly-logarithmic flow-cut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes [21, 22, 13, 19, 12]. Our results from a preliminary version have already found applications in wireless network information flow [16, 7] and we anticipate more in the future. On the technical side our key tools are the formulation and analysis of the dual of the flow relaxations via continuous extensions of submodular functions, in particular the Lovász extension. For directed graphs we rely on a simple yet useful reduction from polymatroidal networks to standard networks. For undirected graphs we rely on the interplay between the Lovász extension of a submodular function and line embeddings with low average distortion introduced by Matoušek and Rabinovich [25]; this connection is inspired by, and generalizes, the work of Feige, Hajiaghayi and Lee [12] on node-capacitated multicommodity flows and cuts. The applicability of embeddings to flow-cut gaps in polymatroidal networks is of independent mathematical interest.