IEEE Journal on Selected Areas in Communications - Special issue on stochastic geometry and random graphs for the analysis and designof wireless networks
Flow-cut gaps for integer and fractional multiflows
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Flow-cut gaps for integer and fractional multiflows
Journal of Combinatorial Theory Series B
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In this paper we present a new bound on the min-cut max-flow ratio for multicommodity flow problems with specified demands. For multicommodity flows, this is a generalization of the well-known relationship between the capacity of a minimum cut and the value of the maximum flow of a single commodity flow problem. For multicommodity flows, capacity of a cut is scaled by the demand that has to cross the cut to obtain the numerator of this ratio. In the denominator, the maximum concurrent flow value is used. Currently, the best known bound for this ratio is proportional to $\log(k)$, where $k$ is the number of origin-destination pairs with positive demand. Our new bound is proportional to $\log(k^*)$, where $k^*$ is the cardinality of the minimum cardinality vertex cover of the demand graph. To obtain this bound, we start with a so-called aggregated commodity formulation of the maximum concurrent flow problem with $k^*$ commodities. We also show a similar bound for the maximum multicommodity flow problem. The new bound is proportional to $\min\{\log(k^*), k^{**}\}$, where $k^{**}$ denotes the size of the of the demand graph.