A new approach to the maximum flow problem
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
A new approach to the maximum-flow problem
Journal of the ACM (JACM)
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
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
Spanning trees with minimum weighted degrees
Information Processing Letters
Additive approximation for bounded degree survivable network design
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Approximating Directed Weighted-Degree Constrained Networks
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Network Design with Weighted Degree Constraints
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Approximating directed weighted-degree constrained networks
Theoretical Computer Science
On generalizations of network design problems with degree bounds
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Operations Research Letters
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Given a graph G and degree bound B on its nodes, the bounded-degree minimum spanning tree (BDMST) problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. This bi-criteria optimization problem generalizes several combinatorial problems, including the Traveling Salesman Path Problem (TSPP). An $(\alpha,\:f(B))$-approximation algorithm for the BDMST problem produces a spanning tree that has maximum degree f(B) and cost within a factor α of the optimal cost. Könemann and Ravi [13,14] give a polynomial-time $({1+\frac{1}{\beta}},\:{bB(1+\beta) + \log_bn})$-approximation algorithm for any b 1, β 0. In a recent paper [2], Chaudhuri et al. improved these results with a $({1},\:{bB+\sqrt{b}\log_bn})$-approximation for any b 1. In this paper, we present a $({1+\frac{1}{\beta}},\:{2B(1+ \beta) + o(B(1+\beta))})$-approximation polynomial-time algorithm. That is, we give the first algorithm that approximates both degree and cost to within a constant factor of the optimal. These results generalize to the case of non-uniform degree bounds. The crux of our solution is an approximation algorithm for the related problem of finding a minimum spanning tree (MST) in which the maximum degree of the nodes is minimized, a problem we call the minimum-degree MST (MDMST) problem. Given a graph G for which the degree of the MDMST solution is $\Delta_{\mbox{\sc{opt}}}$, our algorithm obtains in polynomial time an MST of G of degree at most $2\Delta_{\mbox{\sc{opt}}} + o(\Delta_{\mbox{\sc{opt}}})$. This result improves on a previous result of Fischer [4] that finds an MST of G of degree at most $b\Delta_{\mbox{\sc{opt}}} + \log_bn$ for any b 1, and on the improved quasipolynomial algorithm of [2]. Our algorithm uses the push-relabel framework developed by Goldberg [7] for the maximum flow problem. To our knowledge, this is the first instance of a push-relabel approximation algorithm for an NP-hard problem, and we believe these techniques may have larger impact. We note that for B = 2, our algorithm gives a tree of cost within a (1+ε)-factor of the optimal solution to TSPP and of maximum degree $O(\frac{1}{\epsilon})$ for any ε 0, even on graphs not satisfying the triangle inequality.