### 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.

Original language | English (US) |
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Pages (from-to) | 520-525 |

Number of pages | 6 |

Journal | Journal of Discrete Algorithms |

Volume | 6 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2008 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Approximation
- Flow
- Location
- Source

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## Cite this

*Journal of Discrete Algorithms*,

*6*(3), 520-525. https://doi.org/10.1016/j.jda.2006.12.010