Spanning trees with variable degree bounds

In this paper, we introduce and study a generalization of the degree constrained minimum spanning tree problem where we may install one of several available transmission systems (each with a different cost value) in each edge. The degree of the endnodes of each edge depends on the system installed on the edge. We also discuss a particular case that arises in the design of wireless mesh networks (in this variant the degree of the endnodes of each edge depend on the transmission system installed on it as well as on the length of the edge). We propose three classes of models using different sets of variables and compare from a theoretical perspective as well as from a computational point of view, the models and the corresponding linear programming relaxations. The computational results show that some of the proposed models are able to solve to optimality instances with 100 nodes and different scenarios.

[1]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

[2]  Rajeev Rastogi,et al.  VillageNet: A low-cost, IEEE 802.11-based mesh network for connecting rural areas , 2007, Bell Labs Technical Journal.

[3]  Alexandre Salles da Cunha,et al.  Lower and upper bounds for the degree-constrained minimum spanning tree problem , 2007, Networks.

[4]  Xavier Muñoz,et al.  Graphs and Algorithms in Communication Networks: Studies in Broadband, Optical, Wireless and Ad Hoc Networks , 2010, Texts in Theoretical Computer Science. An EATCS Series.

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  W. Marsden.  I and J , 2012 .

[7]  Pedro Martins,et al.  Min-degree constrained minimum spanning tree problem: complexity, properties, and formulations , 2012, Int. Trans. Oper. Res..

[8]  Luís Gouveia,et al.  Prize collecting Steiner trees with node degree dependent costs , 2011, Comput. Oper. Res..

[9]  Cheng Li,et al.  Channel Assignment Exploiting Partially Overlapping Channels for Wireless Mesh Networks , 2009, GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference.

[10]  Christophe Duhamel,et al.  Models and heuristics for the k ‐degree constrained minimum spanning tree problem with node‐degree costs , 2012, Networks.

[11]  Veljko Pejovic,et al.  Broadband Adoption| The Bandwidth Divide: Obstacles to Efficient Broadband Adoption in Rural Sub-Saharan Africa , 2012 .

[12]  Luís Gouveia,et al.  Spanning Trees with Node Degree Dependent Costs and Knapsack Reformulations , 2010, Electron. Notes Discret. Math..

[13]  Matteo Fischetti,et al.  An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem , 2006, Math. Program..

[14]  Mahesh K. Marina,et al.  Tegola tiered mesh network testbed in rural Scotland , 2008, WiNS-DR '08.

[15]  David Peleg,et al.  Graphs and Algorithms in Communication Networks , 2010 .

[16]  David W. Corne,et al.  A new evolutionary approach to the degree-constrained minimum spanning tree problem , 1999, IEEE Trans. Evol. Comput..

[17]  Rajeev Rastogi,et al.  VillageNet: A low-cost, IEEE 802.11-based mesh network for connecting rural areas , 2007 .

[18]  Di Yuan,et al.  Mathematical Optimization Models for WLAN Planning , 2010, Graphs and Algorithms in Communication Networks.

[19]  Alexandre Salles da Cunha,et al.  Lower and upper bounds for the degree-constrained minimum spanning tree problem , 2007 .

[20]  Falko Dressler,et al.  Realistic simulation and experimental validation of adjacent-channel interference in planning of industrial wireless networks , 2011, PE-WASUN '11.

[21]  Josep Paradells Aspas,et al.  Effect of adjacent-channel interference in IEEE 802.11 WLANs , 2007, 2007 2nd International Conference on Cognitive Radio Oriented Wireless Networks and Communications.

[22]  L. Caccetta,et al.  A branch and cut method for the degree-constrained minimum spanning tree problem , 2001 .

[23]  Peter Eades,et al.  On Optimal Trees , 1981, J. Algorithms.

[24]  Kameswari Chebrolu,et al.  Experiences in using WiFi for rural internet in India , 2007, IEEE Communications Magazine.