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Improved Bi-criteria Approximation for the All-or-Nothing Multicommodity Flow Problem in Arbitrary Networks
This paper addresses the following fundamental maximum throughput routing
problem: Given an arbitrary edge-capacitated -node directed network and a
set of commodities, with source-destination pairs and demands
, admit and route the largest possible number of commodities -- i.e.,
the maximum {\em throughput} -- to satisfy their demands. The main
contributions of this paper are two-fold: First, we present a bi-criteria
approximation algorithm for this all-or-nothing multicommodity flow (ANF)
problem. Our algorithm is the first to achieve a {\em constant approximation of
the maximum throughput} with an {\em edge capacity violation ratio that is at
most logarithmic in }, with high probability. Our approach is based on a
version of randomized rounding that keeps splittable flows, rather than
approximating those via a non-splittable path for each commodity: This allows
our approach to work for {\em arbitrary directed edge-capacitated graphs},
unlike most of the prior work on the ANF problem. Our algorithm also works if
we consider the weighted throughput, where the benefit gained by fully
satisfying the demand for commodity is determined by a given weight
. Second, we present a derandomization of our algorithm that maintains
the same approximation bounds, using novel pessimistic estimators for
Bernstein's inequality. In addition, we show how our framework can be adapted
to achieve a polylogarithmic fraction of the maximum throughput while
maintaining a constant edge capacity violation, if the network capacity is
large enough. One important aspect of our randomized and derandomized
algorithms is their {\em simplicity}, which lends to efficient implementations
in practice