## Abstract

Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edge-cost-flow problems (a.k.a. fixed-charge flows). We prove a hardness result for the Minimum Edge Cost Flow Problem (MECF). Using the one-round two-prover scenario, we prove that MECF in directed graphs does not admit a 2^{log1−ε} ^{n}-ratio approximation, for every constant ε > 0, unless NP ⊆ DTIME(n^{polylogn}). A restricted version of MECF, called Infinite Capacity MECF (ICF), is defined as follows: (i) all edges have infinite capacity, (ii) there are multiple sources and sinks, where flow can be delivered from every source to every sink, (iii) each source and sink has a supply amount and demand amount, respectively, and (iv) the required total flow is given as part of the input. The goal is to find a minimum edgecost flow that meets the required total flow while obeying the demands of the sinks and the supplies of the sources. We prove that directed ICF generalizes the Covering Steiner Problem. We also show that the undirected version of ICF generalizes several network design problems, such as: Steiner Tree Problem, k-MST, Point-to-point Connection Problem, and the generalized Steiner Tree Problem. An O(log x)-approximation algorithm for undirected ICF is presented, where x denotes the required total flow. We also present a bi-criteria approximation algorithm for directed ICF. The algorithm for directed ICF finds a flow that delivers half the required flow at a cost that is at most O(n^{ε}/ε^{5}) times bigger than the cost of an optimal flow. The running time of the algorithm for directed ICF is O(x^{2/ε} ・ n^{1+1/ε}). Finally, randomized approximation algorithms for the Covering Steiner Problem in directed and undirected graphs are presented. The algorithms are based on a randomized reduction to a problem called 12 -Group Steiner. This reduction can be derandomized to yield a deterministic reduction. In directed graphs, the reduction leads to a first non-trivial approximation algorithm for the Covering Steiner Problem. In undirected graphs, the resulting ratio matches the best ratio known [KRS01], via a much simpler algorithm.

Original language | English |
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Title of host publication | Algorithm Theory - SWAT 2002 - 8th Scandinavian Workshop on Algorithm Theory, Proceedings |

Editors | Martti Penttonen, Erik Meineche Schmidt |

Publisher | Springer Verlag |

Pages | 318-327 |

Number of pages | 10 |

ISBN (Print) | 9783540438663 |

DOIs | |

State | Published - 2002 |

Externally published | Yes |

Event | 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002 - Turku, Finland Duration: 3 Jul 2002 → 5 Jul 2002 |

### 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 | 2368 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002 |
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Country/Territory | Finland |

City | Turku |

Period | 3/07/02 → 5/07/02 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 2002.