639 research outputs found
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 c > 0, if the node
cut conditions are satisfied, then one can simultaneously route a c-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.Comment: 30 pages, 5 figure
Single Source - All Sinks Max Flows in Planar Digraphs
Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing
max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution was known than running a single-source single-sink max flow
algorithm n-1 times, giving a total time bound of O(n^2 log n) with the
algorithm of Borradaile and Klein.
An important implication is that all-pairs max st-flow values in G can be
computed in near-quadratic time. This is close to optimal as the output size is
Theta(n^2). We give a quadratic lower bound on the number of distinct max flow
values and an Omega(n^3) lower bound for the total size of all min cut-sets.
This distinguishes the problem from the undirected case where the number of
distinct max flow values is O(n).
Previous to our result, no algorithm which could solve the all-pairs max flow
values problem faster than the time of Theta(n^2) max-flow computations for
every planar digraph was known.
This result is accompanied with a data structure that reports min cut-sets.
For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
- …