Abstract
For a given undirected graph with each edge associated with a weight and a length, the constrained minimum spanning tree (CMST) problem aims to compute a minimum weight spanning tree with total length bounded by a given fixed integer \(L\in {\mathbb {Z}}^{+}\). In the paper, we first present an exact algorithm with a runtime \(O(mn^{2})\) for CMST when the edge length is restricted to 0 and 1 based on combining the local search method and our developed bicameral edge replacement approach. Then we extend the algorithm to solve a more general case when the edge length is restricted to 0, 1 and 2 via iteratively improving a feasible solution of CMST towards an optimum solution. At last, numerical experiments are carried out to validate the practical performance of the proposed algorithms by comparing with previous algorithms as baselines.
Similar content being viewed by others
References
Aggarwal V, Aneja YP, Nair KPK (1982) Minimal spanning tree subject to a side constraint. Comput Oper Res 9(4):287–296
Alfandari L, Paschos VT (1999) Approximating minimum spanning tree of depth 2. Int Trans Oper Res 6(6):607–622
Bicalho LH, Cunha ASD, Lucena A (2016) Branch-and-cut-and-price algorithms for the degree constrained minimum spanning tree problem. Comput Optim Appl 63(3):1–38
Boldon B, Deo N, Kumar N (1996) Minimum-weight degree-constrained spanning tree problem: Heuristics and implementation on an simd parallel machine. Parallel Comput 22(3):369–382
Dahl G (1998) The 2-hop spanning tree problem. Oper Res Lett 23(1):21–26
Gao X, Jia L (2016) Degree-constrained minimum spanning tree problem with uncertain edge weights. Appl Soft Comput 56:580–588
Guo L, Liao K, Shen H, Li P (2015) Brief announcement: Efficient approximation algorithms for computing k disjoint restricted shortest paths. In: Proceedings of the 27th ACM on symposium on parallelism in algorithms and architectures, SPAA 2015, Portland, OR, USA, June 13–15, 2015, pp 62–64
Hassin R, Levin A (2004) An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection. SIAM J Comput 33(2):261–268
Hong SP, Chung SJ, Park BH (2004) A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. Oper Res Lett 32(3):233–239
Korte B, Vygen J, Korte B, Vygen J (2002) Combinatorial optimization, vol 1. Springer, New York
Kruskal JB (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proc Am Math Soc 7(1):48–50
Lau LC, Singh M (2007) Iterative rounding and relaxation (combinatorial). ACM Symp Theory Comput (STOC) 651:660
Marathe M, Ravi R, Ravi S, Rosenkrantz D, Hunt H (1995) Bicriteria network design problems. In: Automata, languages and programming, pp 487–498
Narula SC, Ho CA (1980) Degree-constrained minimum spanning tree. Comput Oper Res 7(4):239–249
Prim RC (1957) Shortest connection networks and some generalizations. Bell Labs Tech J 36(6):1389–1401
Ravi R, Goemans M (1996) The constrained minimum spanning tree problem. In: Algorithm theory SWAT’96, pp 66–75
Acknowledgements
The research is supported by National Science Foundation of China (No. 61772005), Innovative Team of Youth and Creative Science and Technology Program of Shandong Province (2020KJN008) and Natural Science Foundation of Fujian Province (No. 2017J01753).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dedicated to Professor Minyi Yue on the Occasion of His 100th Birthday.
Rights and permissions
About this article
Cite this article
Yao, P., Guo, L. Exact algorithms for finding constrained minimum spanning trees. J Comb Optim 44, 2085–2103 (2022). https://doi.org/10.1007/s10878-020-00579-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-020-00579-z