Mohammad Moharrami

A node-capacitated Okamura---Seymour theorem

The classical Okamura–Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal

Dimension Reduction for Finite Trees in \varvec{\ell _1}

\varepsilon > 0

Robust Algorithms for on Minor-Free Graphs Based on the Sherali-Adams Hierarchy

ε>0, if the node cut conditions are satisfied, then one can simultaneously route an

Bilipschitz snowflakes and metrics of negative type

\varepsilon

Dimension reduction for finite trees in l 1

ε-fraction of all the demands. This answers a question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of the multi-commodity polymatroid networks introduced by Chekuri et al. (ITCS, pp 399–408, 2012). In their framework, one associates to each node a submodular function on the adjacent edges that dictates the types of flows the node can support. In order to round the convex programs corresponding to node and polymatroid-capacitated flows, we devise a new type of random metric embedding that preserves some of the combinatorial structure of the underlying graph.

On the nonexistence of dimension reduction for

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal ε > 0, if the node cut conditions are satisfied, then one can simultaneously route an ε-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.