Theory of Computing
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Title : Single Source Multiroute Flows and Cuts on Uniform Capacity Networks
Authors : Henning Bruhn, Jakub Cerny, Alexander Hall, Petr Kolman, and Jiri Sgall
Volume : 4
Number : 1
Pages : 1-20
URL : https://theoryofcomputing.org/articles/v004a001
Abstract
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For an integer h > 0, an *elementary h-route flow* is a flow
along h edge disjoint paths between a source and a sink, each
path carrying a unit of flow, and a single commodity h-route flow
is a non-negative linear combination of elementary h-route flows.
An instance of a *single source multicommodity flow problem* for
a graph G=(V,E) consists of a source vertex s\in V and k sinks
t_1,...,t_k \in V corresponding to k commodities; we denote it
I = (s;t_1,...,t_k). In the *single source multicommodity multiroute
flow problem*, we are given an instance I = (s;t_1,...,t_k) and an
integer h > 0, and the objective is to maximize the total amount of
flow that is transferred from the source to the sinks so that the
capacity constraints are obeyed and, moreover, the flow of each
commodity is an h-route flow.
We study the relation between classical and multiroute single source
flows on undirected networks with uniform capacities and we provide
a tight bound. In particular, we prove the following result. Given
an instance I = (s;t_1,...,t_k) such that each s-t_i pair is
h-connected, the maximum classical flow between s and the t_i
is at most (2-2/h)-times larger than the maximum h-route flow
between s and the t_i and this is the best possible bound
for h > 1. This, as we show, is in contrast to the situation of
general multicommodity (i.e., multiple sources or non-uniform
capacities) multiroute flows that are up to k(1-1/h)-times smaller
than their classical counterparts.
Furthermore, we introduce and investigate *duplex flows* defined
so that the capacity constraints on edges are applied
independently to each direction. We show that for networks with
uniform capacities and for instances as above the maximum classical
flow between s and the t_i is the same as the maximum h-route
duplex flow between $s$ and $t_i$'s. Moreover, the total flow on
each edge in the duplex flow can be restricted to (2-2/h)C, where
C is the capacity of each edge.
As a corollary, we establish a max-flow min-cut theorem for the
single source multicommodity multiroute flow and cut. An
h-disconnecting cut for I is a set of edges F\subseteq E such
that for each i, the maximum h-route flow between s and t_i
is zero. We show that the maximum h-route flow is within 2h-2
of the minimum h-disconnecting cut, independently of the number
of commodities; we also describe a (2h-2)-approximation algorithm
for the minimum h-disconnecting cut problem.