Min-degree constrained minimum spanning tree problem with fixed centrals and terminals: Complexity, properties and formulations

A variant of the Min-Degree Constrained Minimum Spanning Tree Problem is proposed.Each node is fixed as central (with a minimum degree constraint) or terminal (leaf).The feasibility problem is shown to be NP-Complete.Necessary and sufficient conditions for feasibility are derived.Several ILP formulations and three Lagrangian heuristics are presented and tested. We consider a variant of the Min-Degree Constrained Minimum Spanning Tree Problem where the central and terminal nodes are fixed a priori. We prove that the optimization problem is NP-Hard even for complete graphs and the feasibility problem is NP-Complete even if there is an edge between each central and each terminal in the input graph. Actually, this complexity result still holds when the minimum degree of each central node is restricted to be a same value d 2. We derive necessary and sufficient conditions for feasibility. We present several integer linear programming formulations based on known formulations for the minimum spanning tree problem along with a theoretical comparison among the lower bounds provided by their linear relaxations. We propose three Lagrangian heuristics. Computational experiments compare the performances of the heuristics and the formulations.

[1]  Laurence A. Wolsey,et al.  Trees and Cuts , 1983 .

[2]  Alexandre Salles da Cunha,et al.  The min-degree constrained minimum spanning tree problem: Formulations and Branch-and-cut algorithm , 2014, Discret. Appl. Math..

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

[4]  Douglas S. Reeves,et al.  The delay-constrained minimum spanning tree problem , 1997, Proceedings Second IEEE Symposium on Computer and Communications.

[5]  Pedro Martins,et al.  VNS and second order heuristics for the min-degree constrained minimum spanning tree problem , 2009, Comput. Oper. Res..

[6]  Lou Caccetta,et al.  Computational methods for the diameter restricted minimum weight spanning tree problem , 1994, Australas. J Comb..

[7]  P. Hall.  On Representatives of Subsets , 1935 .

[8]  Barbaros Ç. Tansel,et al.  Min-degree constrained minimum spanning tree problem: New formulation via Miller-Tucker-Zemlin constraints , 2010, Comput. Oper. Res..

[9]  P. Camerini,et al.  On improving relaxation methods by modified gradient techniques , 1975 .

[10]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[11]  Oystein Ore,et al.  On a graph theorem by dirac , 1967 .

[12]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[13]  Celso C. Ribeiro,et al.  Variable neighborhood search for the degree-constrained minimum spanning tree problem , 2002, Discret. Appl. Math..

[14]  R. Kipp Martin,et al.  Using separation algorithms to generate mixed integer model reformulations , 1991, Oper. Res. Lett..

[15]  Pedro Martins,et al.  Skewed VNS enclosing second order algorithm for the degree constrained minimum spanning tree problem , 2008, Eur. J. Oper. Res..

[16]  William J. Cook,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, 50 Years of Integer Programming.

[17]  Luís Gouveia,et al.  Reformulation by Intersection Method on the MST Problem with Lower Bound on the Number of Leaves , 2011, INOC.

[18]  Alexandre Salles da Cunha,et al.  A Parallel Lagrangian Relaxation Algorithm for the Min-Degree Constrained Minimum Spanning Tree Problem , 2012, ISCO.

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

[20]  G. Nemhauser,et al.  Integer Programming , 2020 .

[21]  Abilio Lucena,et al.  Using Lagrangian dual information to generate degree constrained spanning trees , 2006, Discret. Appl. Math..

[22]  Luís Gouveia,et al.  Using the Miller-Tucker-Zemlin constraints to formulate a minimal spanning tree problem with hop constraints , 1995, Comput. Oper. Res..

[23]  Antonio Frangioni,et al.  Convergence Analysis of Deflected Conditional Approximate Subgradient Methods , 2009, SIAM J. Optim..

[24]  L. Wolsey,et al.  Chapter 9 Optimal trees , 1995 .

[25]  Subhash C. Narula,et al.  Degree-constrained minimum spanning tree , 1980, Comput. Oper. Res..

[26]  Gilbert Laporte,et al.  Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints , 1991, Oper. Res. Lett..

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

[28]  Martin Fürer,et al.  Approximating the Minimum-Degree Steiner Tree to within One of Optimal , 1994, J. Algorithms.