Simultaneous source location

We consider the problem of simultaneous source location: selecting locations for sources in a capacitated graph such that a given set of demands can be satisfied simultaneously, with the goal of minimizing the number of locations chosen. For general directed and undirected graphs we give an <i>O</i>(log <i>D</i>)-approximation algorithm, where <i>D</i> is the sum of demands, and prove matching Ω(log <i>D</i>) hardness results assuming <b>P</b> ≠ <b>NP</b>. For undirected trees, we give an exact algorithm and show how this can be combined with a result of Räcke to give a solution that exceeds edge capacities by at most <i>O</i>(log<sup>2</sup> <i>n</i> log log <i>n</i>), where <i>n</i> is the number of nodes. For undirected graphs of bounded treewidth we show that the problem is still <b>NP</b>-hard, but we are able to give a PTAS with at most (1 + &epsis;) violation of the capacities for arbitrarily small &epsis;, or a (<i>k</i>+1) approximation with exact capacities, where <i>k</i> is the treewidth.

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