Abstract
Given an integer \(L\in \mathbb {Z}^{+}\) and an undirected graph with a weight and a length associated with every edge, the constrained minimum spanning tree (CMST) problem is to compute a minimum weight spanning tree with total length bounded by L. The problem was shown weakly \(\mathcal{NP}\)-hard in [1], admitting a PTAS with a runtime \(O(n^{O(\frac{1}{\epsilon })}(m\log ^{2}n+n\log ^{3}n))\) due to Ravi and Goemans [13]. In the paper, we present an exact algorithm for CMST, based on our developed bicameral edge replacement which improves a feasible solution of CMST towards an optimal solution. By applying the classical rounding and scaling technique to the exact algorithm, we can obtain a fully polynomial-time approximation scheme (FPTAS), i.e. an approximation algorithm with a ratio \((1+\epsilon )\) and a runtime \(O(mn^{5}\frac{1}{\epsilon ^{2}})\), where \(\epsilon >0\) is any fixed real number.
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Acknowledgements
The research of the first author is supported by Natural Science Foundation of China (Nos. 61772005, 61300025) and Natural Science Foundation of Fujian Province (No. 2017J01753).
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Yao, P., Guo, L. (2017). Fast Approximation Algorithms for Computing Constrained Minimum Spanning Trees. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10627. Springer, Cham. https://doi.org/10.1007/978-3-319-71150-8_9
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DOI: https://doi.org/10.1007/978-3-319-71150-8_9
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