Abstract
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.
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Chan, T.M.: Euclidean bounded-degree spanning tree ratios. In: Proceedings of the nineteenth annual symposium on Computational geometry, pp. 11–19. ACM Press, New York (2003)
Chaudhuri, K., Rao, S., Riesenfeld, S., Talwar, K.: What would edmonds do? augmenting paths and witnesses for bounded degree msts. In: Proceedings of APPROX/RANDOM (2005)
Edmonds, J.: Maximum matching and a polyhedron with 0–1 vertices. Journal of Research National Bureau of Standards 69B, 125–130 (1965)
Fischer, T.: Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept of Computer Science, Cornell University, Ithaca, NY (1993)
Fürer, M., Raghavachari, B.: Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms 17(3), 409–423 (1994)
Goemans, M.: Personal communication (2006)
Golberg, A.V.: A new max-flow algorithm. Technical Report MIT/LCS/TM-291, Massachussets Institute of Technology, Technical Report (1985)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. In: Proceedings of the eighteenth annual ACM symposium on Theory of computing, pp. 136–146. ACM Press, New York (1986)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)
Hoogeveen, J.A.: Analysis of christofides heuristic: Some paths are more difficult than cycles. Operation Research Letters 10, 291–295 (1991)
Jothi, R., Raghavachari, B.: Degree-bounded minimum spanning trees. In: Proc. 16th Canadian Conf. on Computational Geometry (CCCG) (2004)
Khuller, S., Raghavachari, B., Young, N.: Low-degree spanning trees of small weight. SIAM J. Comput. 25(2), 355–368 (1996)
Könemann, J., Ravi, R.: A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees. SIAM Journal on Computing 31(6), 1783–1793 (2002)
Könemann, J., Ravi, R.: Primal-dual meets local search: approximating MST’s with nonuniform degree bounds. In: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, San Diego, CA, USA, June 9–11, 2003, pp. 389–395. ACM Press, New York (2003)
Papadimitriou, C.H., Vazirani, U.: On two geometric problems related to the traveling salesman problem. J. Algorithms 5, 231–246 (1984)
Ravi, R., Singh, M.: Delegate and conquer: An LP-based approximation algorithm for minimum degree msts. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, Springer, Heidelberg (2006)
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Chaudhuri, K., Rao, S., Riesenfeld, S., Talwar, K. (2006). A Push-Relabel Algorithm for Approximating Degree Bounded MSTs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_18
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DOI: https://doi.org/10.1007/11786986_18
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