## Abstract

In Source Location (SL) problems the goal is to select a minimum cost source set S ⊆ V such that the connectivity (or flow) ψ(S, v) from S to any node v is at least the demand d_{v} of v. In many SL problems ψ(S, v) = d_{v} if v ∈ S, so the demand of nodes selected to S is completely satisfied. In a variant suggested recently by Fukunaga [7], every node v selected to S gets a “bonus” p_{v} ≤ d_{v}, and ψ(S, v) = p_{v} + κ(S \ {v}, v) if v ∈ S and ψ(S, v) = κ(S, v) otherwise, where κ(S, v) is the maximum number of internally disjoint (S, v)-paths. While the approximability of many SL problems was seemingly settled to Θ(ln d(V )) in [20], for his variant on undirected graphs Fukunaga achieved ratio O(k ln k), where k = max_{v}_{∈}V d_{v} is the maximum demand. We improve this by achieving ratio min{p^{∗} ln k, k} · O(ln k) for a more general version with node capacities, where p^{∗} = ma_{xv}∈V pv is the maximum bonus. In particular, for the most natural case p^{∗} = 1 we improve the ratio from O(k ln k) to O(ln^{2} k). To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We obtain ratio O(min{ln n, ln^{2} k}) for this variant, improving over the best ratio known for the general case O(k^{3} ln n) of Chuzhoy and Khanna [3]. In addition, we show that directed SL with unit costs is Ω(log n)-hard to approximate even for 0, 1 demands, while SL with uniform demands can be solved in polynomial time. Finally, we obtain a logarithmic ratio for a generalization of SL where we also have edge-costs and flow-cost bounds {b_{v}: v ∈ V }, and require that the minimum cost of a flow of value d_{v} from S to every node v is at most b_{v}.

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Revised Papers |

Editors | Ernst W. Mayr |

Publisher | Springer Verlag |

Pages | 203-218 |

Number of pages | 16 |

ISBN (Print) | 9783662531730 |

DOIs | |

State | Published - 2016 |

Event | 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015 - Garching, Germany Duration: 17 Jun 2015 → 19 Jun 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9224 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015 |
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Country/Territory | Germany |

City | Garching |

Period | 17/06/15 → 19/06/15 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing Switzerland 2016.