Journal of Global Optimization

, Volume 70, Issue 3, pp 677–685

# Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$-norm

Article

## Abstract

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.

## Keywords

Partial inverse problem Spanning tree Polynomial time algorithm Computational complexity

## Notes

### Acknowledgements

This research is supported by NSFC (Nos. 11571155, 11531011, and 61222201) and the Fundamental Research Funds for the Central Universities (Nos. lzujbky-2017-163 and lzujbky-2016-102).

## References

1. 1.
Ahuja, R.K., Orlin, J.B.: Inverse optimiztion. Oper. Res. 49(5), 771–783 (2001)
2. 2.
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)
3. 3.
Burton, D., Toint, Ph.L.: On an instance of the inverse shortest paths problem. Math. Progr. 53(1), 45–61 (1992)Google Scholar
4. 4.
Cai, M.-C., Duin, C.W., Yang, X., Zhang, J.: The partial inverse minimum spanning tree problem when weight increase is forbidden. Eur. J. Oper. Res. 188, 348–353 (2008)
5. 5.
Cheriyan, J., Hagerup, T., Mehlhorn, K.: An $$O(n^3)$$-time maximum-flow algorithm. SIAM J. Comput. 25(6), 1144–1170 (1996)
6. 6.
Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)
7. 7.
Dell’Amico, M., Maffioli, F., Malucelli, F.: The base-matroid and inverse combinatorial optimization problems. Discrete Appl. Math. 128, 337–353 (2003)
8. 8.
Demange, M., Monnot, J.: An introductuion to inverse combinatorial problems. In: Paschos, V.Th (ed.) Paradigms of Combinatorial Optimization, 2nd edn. Wliey, Hoboken (2014)Google Scholar
9. 9.
Gassner, E.: The partial inverse minimum cut problem with $$L_1$$-norm is strongly NP-hard. RAIRO Oper. Res. 44, 241–249 (2010)
10. 10.
Guan, X., Pardalos, P.M., Zuo, X.: Inverse Max+Sum spanning tree problem by modifying the sum-cost vector under weighted $$l_{\infty }$$ Norm. J. Glob. Optim. 61(1), 165–182 (2015)
11. 11.
Heuberger, C.: Inverse combinatorial optimization: a survey on problems, methods, and results. J. Comb. Optim. 8, 329–361 (2004)
12. 12.
Lai, T., Orlin, J.: The complexity of preprocessing. Research Report of Sloan School of Management, MIT (2003)Google Scholar
13. 13.
Li, S., Zhang, Z., Lai, H.-J.: Algorithm for constraint partial inverse matroid problem with weight increase forbidden. Theor. Comput. Sci. 640, 119–124 (2016)
14. 14.
Orlin, J. B.: Max flows in $$O(nm)$$ time, or better. In: Proceedings of the forty-fifth annual ACM Symposium on Theory of Computing (STOC 2013), 765–774 (2013)Google Scholar
15. 15.
Yang, X.: Complexity of partial inverse assignment problem and partial inverse cut problem. RAIRO Oper. Res. 35, 117–126 (2001)
16. 16.
Yang, X., Zhang, J.: Partial inverse assignment problem under $$l_1$$ norm. Oper. Res. Lett. 35, 23–28 (2007)
17. 17.
Yang, X., Zhang, J.: Inverse sorting problem by minimizing the total weighted number of changers and partial inverse sorting problems. Comput. Optim. Appl. 36(1), 55–66 (2007)
18. 18.
Zhang, Z., Li, S., Lai, H.-J., D, D.-Z.: Algorithms for the partial inverse matroid problem in which weights can only be increased. J. Glob. Optim. 65(4), 801–811 (2016)

## Authors and Affiliations

1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
2. 2.College of Mathematics Physics and Information EngineeringZhejiang Normal UniversityJinhuaPeople’s Republic of China
3. 3.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA