Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
On max-flow min-cut and integral flow properties for multicommodity flows in directed networks
Information Processing Letters
Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
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
Multicommodity flow, well-linked terminals, and routing problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Graph Theory With Applications
Graph Theory With Applications
Embedding k-Outerplanar Graphs into l1
SIAM Journal on Discrete Mathematics
Probabilistic embeddings of bounded genus graphs into planar graphs
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
A New Min-Cut Max-Flow Ratio for Multicommodity Flows
SIAM Journal on Discrete Mathematics
Hardness of the Undirected Congestion Minimization Problem
SIAM Journal on Computing
On partitioning graphs via single commodity flows
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Coarse Differentiation and Multi-flows in Planar Graphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
On the geometry of graphs with a forbidden minor
Proceedings of the forty-first annual ACM symposium on Theory of computing
Graph partitioning using single commodity flows
Journal of the ACM (JACM)
Randomly removing g handles at once
Proceedings of the twenty-fifth annual symposium on Computational geometry
A Note on Multiflows and Treewidth
Algorithmica
Edge disjoint paths in moderately connected graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Approximating sparsest cut in graphs of bounded treewidth
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Sparsest cut on bounded treewidth graphs: algorithms and hardness results
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
Pathwidth, trees, and random embeddings
Combinatorica
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Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flow-cut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C. It is well-known that the flow-cut gap may be greater than 1 even in the case where G is the (series-parallel) graph K2, 3. In this paper we are primarily interested in the "integer" flow-cut gap. What is the minimum value C such that there exists a feasible integer valued multiflow for H if each edge of G is given capacity C? We formulate a conjecture that states that the integer flow-cut gap is quantitatively related to the fractional flow-cut gap. In particular this strengthens the well-known conjecture that the flow-cut gap in planar and minor-free graphs is O(1) [12] to suggest that the integer flow-cut gap is O(1). We give several technical tools and results on non-trival special classes of graphs to give evidence for the conjecture and further explore the "primal" method for understanding flow-cut gaps; this is in contrast to and orthogonal to the highly successful metric embeddings approach. Our results include the following: • Let G be obtained by series-parallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed path in the resulting oriented graph. We show that if the cut condition holds for a compliant instance and G + H is Eulerian, then an integral routing of H exists. This result includes, as a special case, routing on a ring, but is not a special case of the Okamura-Seymour theorem. • Using the above result, we show that the integer flow-cut gap in series-parallel graphs is 5. • The integer flow-cut gap in k-Outerplanar graphs is cO(k) for some fixed constant c. • A simple proof that the flow-cut gap is O(log k*) where k* is the size of a node-cover in H; this was previously shown by Günlük via a more intricate proof [11].