## Abstract

We consider the directed Min-Cost Rooted Subset k -Edge-Connection problem: given a digraph G= (V, E) with edge costs, a set T⊆ V of terminals, a root node r, and an integer k, find a min-cost subgraph of G that contains k edge disjoint rt-paths for all t∈ T . The case when every edge of positive cost has head in T admits a polynomial time algorithm due to Frank (Discret Appl Math 157(6):1242–1254, 2009), and the case when all positive cost edges are incident to r is equivalent to the k -Multicover problem. Chan et al. (APPROX/RANDOM, 2020) gave an LP-based O(ln kln | T|) -approximation algorithm for quasi-bipartite instances, when every edge in G has at least one end in T∪ { r} . We give a simple combinatorial algorithm with the same approximation ratio for a more general problem of covering an arbitrary T-intersecting supermodular set function by a min-cost edge set, and for the case when only every positive cost edge has at least one end in T∪ { r} .

Original language | English |
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Article number | 10 |

Journal | Operations Research Forum |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2024 |

### Bibliographical note

Publisher Copyright:© 2024, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

## Keywords

- Approximation algorithms
- Min-cost rooted k-edge-connection
- Quasi-bipartite digraphs
- T-intersecting supermodular set functions