## Abstract

Given a graph H = (V,F) with edge weights {w(e):e ∈ F}, the weighted degree of a node v in H is ∑ {w(vu):vu ∈ F}. We give bicriteria approximation algorithms for problems that seek to find a minimum cost directed graph that satisfies both intersecting supermodular connectivity requirements and weighted degree constraints. The input to such problems is a directed graph G = (V,E), edge-costs {c(e):e ∈ E}, edge-weights {w(e):e ∈ E}, an intersecting supermodular set-function f on V, and degree bounds {b(v):v ∈ V}. The goal is to find a minimum cost f-connected subgraph H = (V,F) (namely, at least f(S) edges in F enter every S ⊆ V) of G with weighted degrees ≤ b(v). Our algorithm computes a solution of cost ≤ 2, so that the weighted degree of every v ∈ V is at most: 7 b(v) for arbitrary f and 5 b(v) for a 0,1-valued f; 2b(v) + 4 for arbitrary f and 2b(v) + 2 for a 0,1-valued f in the case of unit weights. Another algorithm computes a solution of cost ≤ 3.opt and weighted degrees ≤ 6b(v). We obtain similar results when there are both indegree and outdegree constraints, and better results when there are indegree constraints only: a (1,4)-approximation algorithm for arbitrary weights and a polynomial time algorithm for unit weights. Finally, we consider the problem of packing maximum number k of edge-disjoint arborescences so that their union satisfies weighted degree constraints, and give an algorithm that computes a solution of value at least [k/36].

Original language | English |
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Title of host publication | Approximation, Randomization and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings |

Pages | 219-232 |

Number of pages | 14 |

DOIs | |

State | Published - 2008 |

Event | 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 - Boston, MA, United States Duration: 25 Aug 2008 → 27 Aug 2008 |

### 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 | 5171 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 |
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Country/Territory | United States |

City | Boston, MA |

Period | 25/08/08 → 27/08/08 |