  Motivated by scheduling in Geo-distributed data analysis, we propose a target
location problem for multi-commodity flow (LoMuF for short). Given commodities
to be sent from their resources, LoMuF aims at locating their targets so that
the multi-commodity flow is optimized in some sense. LoMuF is a combination of
two fundamental problems, namely, the facility location problem and the network
flow problem. We study the hardness and algorithmic issues of the problem in
various settings. The findings lie in three aspects. First, a series of
NP-hardness and APX-hardness results are obtained, uncovering the inherent
difficulty in solving this problem. Second, we propose an approximation
algorithm for general undirected networks and an exact algorithm for undirected
trees, which naturally induce efficient approximation algorithms on directed
networks. Third, we observe separations between directed networks and
undirected ones, indicating that imposing direction on edges makes the problem
strictly harder. These results show the richness of the problem and pave the
way to further studies.
