Volume 12, no. 1Pages 150 - 155

On the Existence of an Integer Solution of the Relaxed Weber Problem for a Tree Network

A.V. Panyukov
The problem of finding the optimal arrangement of vertices of a tree network in the installation space representing a finite set is considered. The criterion of optimality is the minimization of the total cost of deployment and the cost of communications. Placement of different tree vertices in one point of the installation space is allowed. This problem is known as Weber problem for a tree network. The statement of Weber problem as an integer linear programming problem is given in this research. It's proved that a set of optimal solutions of corresponding relaxed Weber problem for a tree-network contains the integer solution. This fact allows to prove the existence a saddle point while proving the performance of decomposition algorithms for problems different from problems because of additional restrictions.
Full text
shape allocation problem; linear programming; duality; relaxation; integer solution; polynomial algorithm; Weber problem.
1. Beresnev V.L., Diskretnye zadachi razmeshcheniya i polinomy ot bulevykh peremennykh [Discrete Location Problems and Polynomials of Boolean Variables]. Novosibirsk, Sobolev Institute Press, 2005. (in Russian)
2. Nickel S., Puerto J. Location Theory. Heidelberg, Springer, 2005.
3. Zabudskii G.G., Veremchuk N.S. An Algorithm for Finding an Approximate Solution to the Weber Problem on a Line with Forbidden Gaps. Journal of Applied and Industrial Mathematics, 2016, vol. 10, no. 1, pp. 136-144. DOI: 10.1134/S1990478916010154
4. Zabudskii G.G., Koval A.A. Solving a Maximin Location Problem on the Plane with Given Accuracy. Automation and Remote Control, 2014, vol. 75, pp. 1221-1230. DOI:10.1134/S0005117914070042
5. Panyukov A.V., Pelzwerger B.V. Polynomial Algorithms to Finite Weber Problem for a Tree Network. Journal of Computational and Applied Mathematics, 1991, vol. 35, pp. 291-296. DOI:10.1016/0377-0427(91)90215-6
6. Ivanko E.E. Iterative Equitable Partition of Graph as a Model of Constant Structure Discrete Time Closed Semantic System. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 4, pp. 26-34. DOI: 10.14529/mmp170403
7. Panyukov A.V. The Relaxation Polyhedron of Weber Problem. Non-Smooth and Discontinuous Problems of Control and Optimization, Chelyabinsk, 1998, pp. 171-174.
8. Panyukov A.V. Location of a Tree Network for a Finite Set. Abstracts of the Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications, Kosice, Safary University, 2013, p. 64.