A note on two source location problems

We consider Source Location (SL) problems: given a capacitated network G = (V,E), cost c(v) and a demand d(v) for every v ∈ V , choose a min-cost S ⊆ V so that λ(v, S) ≥ d(v) holds for every v ∈ V , where λ(v, S) is the maximum flow value from v to S. In the directed variant, we have demands d(v) and d(v) and we require λ(S, v) ≥ d(v) and λ(v, S) ≥ d(v). Undirected SL is (weakly) NP-hard on stars with r(v) = 0 for all v except the center. But, it is known to be polynomially solvable for uniform costs and uniform demands. For general instances, both directed an undirected SL admit a (lnD+1)-approximation algorithms, where D is the sum of the demands; up to constant this is tight, unless P=NP. We give a pseudopolynomial algorithm for undirected SL on trees with running time O(|V |∆3), where ∆ = maxv∈V d(v). This algorithm is used to derive a linear time algorithm for undirected SL with ∆ ≤ 3. We also consider the Single Assignment Source Location (SASL) where every v ∈ V should be assigned to a single node s(v) ∈ S. While the undirected SASL is in P, we give a (ln |V |+ 1)-approximation algorithm for the directed case, and show that this is tight, unless P=NP.