A new approach to the maximum flow problem
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Approximating the minimum-degree Steiner tree to within one of optimal
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
Low-Degree Spanning Trees of Small Weight
SIAM Journal on Computing
A Matter of Degree: Improved Approximation Algorithms for Degree-Bounded Minimum Spanning Trees
SIAM Journal on Computing
Euclidean bounded-degree spanning tree ratios
Proceedings of the nineteenth annual symposium on Computational geometry
Primal-dual meets local search: approximating MST's with nonuniform degree bounds
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Minimum Bounded Degree Spanning Trees
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
Approximating minimum bounded degree spanning trees to within one of optimal
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
Additive guarantees for degree bounded directed network design
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Degree bounded matroids and submodular flows
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
What would edmonds do? augmenting paths and witnesses for degree-bounded MSTs
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Delegate and conquer: an LP-based approximation algorithm for minimum degree MSTs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
A push-relabel algorithm for approximating degree bounded MSTs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Analysis of Christofides' heuristic: Some paths are more difficult than cycles
Operations Research Letters
Degree bounded matroids and submodular flows
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Matroidal degree-bounded minimum spanning trees
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Chain-Constrained spanning trees
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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In the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T, such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2@D"o"p"t" (G)+o(@D"o"p"t" (G)), where @D"o"p"t" (G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [T. Fischer, Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept. of Computer Science, Cornell University, Ithaca, NY, 1993] that finds an MST of degree at most b@D"o"p"t" (G)+log"bn, for any b1. The MDMST problem is a special case of the following problem: given a k-ary hypergraph G=(V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T of M such that the degree of T in G is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k^2@D"o"p"t" (G,M)+O(kk@D"o"p"t" (G,M)). We use the push-relabel framework developed by Goldberg [A. V. Goldberg, A new max-flow algorithm, Technical Report MIT/LCS/TM-291, Massachusetts Institute of Technology, 1985 (Technical Report)] for the maximum-flow problem. To our knowledge, this is the first use of the push-relabel technique in an approximation algorithm for an NP-hard problem. The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Konemann and Ravi [J. Konemann, R. Ravi, A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees, SIAM Journal on Computing 31(6) (2002) 1783-1793; J. Konemann, R. Ravi, Primal-dual meets local search: Approximating MST's with nonuniform degree bounds, in: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, 2003, pp. 389-395] and by Chaudhuri et al. [K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar, What would Edmonds do? Augmenting paths and witnesses for bounded degree MSTs, in: Proceedings of APPROX/RANDOM, 2005, pp. 26-39] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimum. These results generalize to the case of nonuniform degree bounds.