We consider the Minimum Multi-Commodity Flow Subgraph (MMCFS) problem: given
a directed graph $G$ with edge capacities $\mathit{cap}$ and a retention ratio
$\alpha\in(0,1)$, find an edge-wise minimum subgraph $G’ \subseteq G$ such that
for all traffic matrices $T$ routable in $G$ using a multi-commodity flow,
$\alpha\cdot T$ is routable in $G’$. This natural yet novel problem is
motivated by recent research that investigates how the power consumption in
backbone computer networks can be reduced by turning off connections during
times of low demand without compromising the quality of service. Since the
actual traffic demands are generally not known beforehand, our approach must be
traffic-oblivious, c'est-à-dire, work for all possible sets of simultaneously routable
traffic demands in the original network.
In this paper we present the problem, relate it to other known problems in
literature, and show several structural results, including a reformulation,
maximum possible deviations from the optimum, and NP-hardness (as well as a
certain inapproximability) already on very restricted instances. The most
significant contribution is a tight $\max(\frac{1}{\alpha}, 2)$-approximation
based on an algorithmically surprisingly simple LP-rounding scheme.

Cet article explore les excursions dans le temps et leurs implications.

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