A note on two source location problems

Abstract

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 in ( v ) and d out ( v ) and we require λ ( S , v ) ⩾ d in ( v ) and λ ( v , S ) ⩾ d out ( 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 ( ln D + 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 | Δ 3 ) , where Δ = max v ∈ V d ( v ) . This algorithm is used to derive a linear time algorithm for undirected SL with Δ ⩽ 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.