An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
The edge-disjoint path problem is NP-complete for series-parallel graphs
Discrete Applied Mathematics - Special issue on selected papers from First Japanese-Hungarian Symposium for Discrete Mathematics and its Applications
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
A New Min-Cut Max-Flow Ratio for Multicommodity Flows
SIAM Journal on Discrete Mathematics
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
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
Flow-cut gaps for integer and fractional multiflows
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Flow-cut gaps for integer and fractional multiflows
Journal of Combinatorial Theory Series B
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Let G=(V,E) be a supply graph and H=(V,F) a demand graph defined on the same set of vertices. An assignment of capacities to the edges of G and demands to the edges of H is said to satisfy the cut condition if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair (G,H) is called cut-sufficient if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on G. We prove a previous conjecture, which states that when the supply graph G is series-parallel, the pair (G,H) is cut-sufficient if and only if (G,H) does not contain an odd spindle as a minor; that is, if it is impossible to contract edges of G and delete edges of G and H so that G becomes the complete bipartite graph K2,p, with p ≥ 3 odd, and H is composed of a cycle connecting the p vertices of degree 2, and an edge connecting the two vertices of degree p. We further prove that if the instance is Eulerian --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.