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FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Survivable network design with degree or order constraints
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
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Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Additive approximation for bounded degree survivable network design
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Network Design with Weighted Degree Constraints
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Additive Guarantees for Degree-Bounded Directed Network Design
SIAM Journal on Computing
Degree bounded forest covering
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Matroidal degree-bounded minimum spanning trees
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
On generalizations of network design problems with degree bounds
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Network design with weighted degree constraints
Discrete Optimization
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We consider two related problems, the MINIMUM BOUNDED DEGREE MATROID BASIS problem and the MINIMUM BOUNDED DEGREE SUBMODULAR FLOW problem. The first problem is a generalization of the MINIMUM BOUNDED DEGREE SPANNING TREE problem: we are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e) ≤ g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a sub-modular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, and the task is to find a minimum cost 0-1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v is between f(v) and g(v). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e) - 2Δ + 1 ≤ |B ∩ e| ≤ g(e) + 2Δ - 1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ - 1. For the second problem we can find a 0-1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v) - 1 and g(v) + 1. These results can be applied to obtain approximation algorithms for different combinatorial optimization problems with degree constraints, including the Minimum CROSSING SPANNING TREE problem, the MINIMUM BOUNDED DEGREE SPANNING TREE UNION problem, the MINIMUM BOUNDED DEGREE DIRECTED CUT COVER problem, and the MINIMUM BOUNDED DEGREE GRAPH ORIENTATION problem.