Acta Mathematica Vietnamica

, Volume 44, Issue 4, pp 813–831

Combinatorial Algorithms for the Uniform-Cost Inverse 1-Center Problem on Weighted Trees

Article

Abstract

Inverse 1-center problem on a network is to modify the edge lengths or vertex weights within certain bounds so that the prespecified vertex becomes an (absolute) 1-center of the perturbed network and the modifying cost is minimized. This paper focuses on the inverse 1-center problem on a weighted tree with uniform cost of edge length modification, a generalization for the analogous problem on an unweighted tree (Alizadeh and Burkard, Discrete Appl. Math. 159, 706–716, 2011). To solve this problem, we first deal with the weighted distance reduction problem on a weighted tree. Then, the weighted distances balancing problem on two rooted trees is introduced and efficiently solved. Combining these two problems, we derive a combinatorial algorithm with complexity of $$O(n^{2})$$ to solve the inverse 1-center problem on a weighted tree if there exists no topology change during the edge length modification. Here, n is the number of vertices in the tree. Dropping this condition, the problem is solvable in $$O(n^{2}\mathbf {c})$$ time, where $$\mathbf {c}$$ is the compressed depth of the tree. Finally, some special cases of the problem with improved complexity, say linear time, are also discussed.

Keywords

Location problem Inverse optimization problem 1-Center problem Tree

Mathematics Subject Classification (2010)

90B10 90B80 90C27

References

1. 1.
Alizadeh, B., Burkard, R.E.: Combinatorial algorithms for inverse absolute and vertex 1-center location problems on trees. Networks 58(3), 190–200 (2011)
2. 2.
Alizadeh, B., Burkard, R.E.: Uniform-cost inverse absolute and vertex center location problems with edge length variations on trees. Discrete Appl. Math. 159(8), 706–716 (2011)
3. 3.
Alizadeh, B., Burkard, R.E., Pferschy, U.: Inverse 1-center location problems with edge length augmentation on trees. Computing 86(4), 331–343 (2009)
4. 4.
Balas, E., Zemel, E.: An algorithm for large zero-one knapsack problems. Oper. Res. 28(5), 1130–1154 (1980)
5. 5.
Baroughi Bonab, F., Burkard, R.E., Gassner, E.: Inverse p-median problems with variable edge lengths. Math. Methods Oper. Res. 73(2), 263–280 (2011)
6. 6.
Baroughi Bonab, F., Burkard, R.E., Alizadeh, B.: Inverse median location problems with variable coordinates. CEJOR Cent. Eur. J. Oper. Res. 18(3), 365–381 (2010)
7. 7.
Burkard, R.E., Pleschiutschnig, C., Zhang, J.: Inverse median problems. Discrete Optim. 1(1), 23–39 (2004)
8. 8.
Burkard, R.E., Pleschiutschnig, C., Zhang, J.: The inverse 1-median problem on a cycle. Discrete Optim. 5(2), 242–253 (2008)
9. 9.
Burkard, R.E., Galavii, M., Gassnner, E.: The inverse Fermat-Weber problem. European J. Oper. Res. 206(1), 11–17 (2010)
10. 10.
Cai, M.C., Yang, X.G., Zhang, J.Z.: The complexity analysis of inverse center location problem. J. Global Optim. 15(2), 213–218 (1999)
11. 11.
Drezner, Z., Hamacher, H.W.: Facility Location: Applications and Theory. Springer, Berlin (2002)
12. 12.
Eiselt, H.A., Marianov, V.: Foundations of location analysis. international series in operations research and management science. Springer, New York (2011)
13. 13.
Galavii, M.: The inverse 1-median problem on a tree and on a path. Electron. Notes Discrete Math. 36, 1241–1248 (2010)
14. 14.
Guan, X., Zhang, B.: Inverse 1-median problem on trees under weighted Hamming distance. J. Global Optim. 54(1), 75–82 (2012)
15. 15.
Hatzl, J., Karrenbauer, A.: A combinatorial algorithm for the 1-median problem in $$\mathbb {R}^{d}$$ with the Chebyshev norm. Oper. Res. Lett. 38(5), 383–385 (2010)
16. 16.
Handler, G.Y.: Minimax location of a facility in an undirected tree graph. Transp. Sci. 7, 287–293 (1973)
17. 17.
Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. I: the p-centers. SIAM J. Appl. Math. 37(3), 513–538 (1979)
18. 18.
Love, R.F., Morris, J.G., Wesolowsky, G.O.: Facilities Location: Models and Methods. North-Holland (1988)Google Scholar
19. 19.
Megiddo, N.: Linear-time algorithms for linear programming in $$\mathbb {R}^{3}$$ and related problems. SIAM J. Comput. 12(4), 759–776 (1983)
20. 20.
Nguyen, K.T.: Inverse 1-median problem on block graphs with variable vertex weights. J. Optim. Theory Appl. 68(3), 944–957 (2016)
21. 21.
Nguyen, K.T.: Reverse 1-center problem on weighted trees. Optimization 65(1), 253–264 (2016)
22. 22.
Nguyen, K.T., Anh, L.Q.: Inverse k-centrum problem on trees with variable vertex weights. Math. Methods Oper. Res. 82(1), 19–30 (2015)
23. 23.
Nguyen, K.T., Chassein, A.: Inverse eccentric vertex problem on networks. CEJOR Cent. Eur. J. Oper. Res. 23(3), 687–698 (2014)
24. 24.
Nguyen, K.T., Sepasian, A.R.: The inverse 1-center problem on trees with variable edge lengths under Chebyshev norm and Hamming distance. J. Comb. Optim. 32(3), 872–884 (2016)
25. 25.
Sepasian, A.R., Rahbarnia, F.: An $$O(n\log n)$$ algorithm for the inverse 1-median problem on trees with variable vertex weights and edge reductions. Optimization 64(3), 595–602 (2013)

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

• Kien Trung Nguyen
• 1
• Huong Nguyen-Thu
• 1
• Nguyen Thanh Hung
• 1
1. 1.Department of Mathematics, Teacher CollegeCan Tho UniversityCan ThoVietnam

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