Preprocessing for a map sectorization problem by means of mathematical programming

The sectorization problem is a particular case of partitioning problems occurring in cartography. The aim is to partition a territory into sectors such that the statistical activity measure of each sector is as close as possible to a given target value. We model this as a problem of minimizing the maximum deviation among all the sectors between their activity measure and their target value. We propose a mathematical programming formulation for the problem, we add some valid inequalities to restrict the solution space and develop a preprocessing procedure to reduce the number of variables. Computational results on different maps highlight the strong efficiency of this reduction procedure.

[1]  G. Nemhauser,et al.  Optimal Political Districting by Implicit Enumeration Techniques , 1970 .

[2]  Pierre Hansen,et al.  Cluster analysis and mathematical programming , 1997, Math. Program..

[3]  Justo Puerto,et al.  The ordered capacitated facility location problem , 2010 .

[4]  Gilbert Laporte,et al.  A tabu search heuristic and adaptive memory procedure for political districting , 2003, Eur. J. Oper. Res..

[5]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[6]  Anuj Mehrotra.  Constrained graph partitioning: decomposition, polyhedral structure and algorithms , 1992 .

[7]  Tony J. Van Roy,et al.  A Cross Decomposition Algorithm for Capacitated Facility Location , 1986, Oper. Res..

[8]  G. Nemhauser,et al.  An Optimization Based Heuristic for Political Districting , 1998 .

[9]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[10]  Geoffrey C. Fox,et al.  Fast and parallel mapping algorithms for irregular problems , 1996, The Journal of Supercomputing.

[11]  S.,et al.  An Efficient Heuristic Procedure for Partitioning Graphs , 2022 .

[12]  Zvi Drezner,et al.  Facility location - applications and theory , 2001 .

[13]  Ting-Lan. Ji.  Evaluation and optimization of image displays. , 1993 .

[14]  Vincent T'Kindt,et al.  Sequencing a single machine with due dates and deadlines: an ILP-based approach to solve very large instances , 2010, J. Sched..

[15]  J. Weaver,et al.  Nonpartisan Political Redistricting by Computer , 1965 .

[16]  李幼升,et al.  Ph , 1989 .

[17]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[18]  Federico Della Croce,et al.  Enumeration of Pareto Optima for a Flowshop Scheduling Problem with Two Criteria , 2007, INFORMS J. Comput..

[19]  Gilbert Laporte,et al.  A comparative analysis of several formulations for the generalized minimum spanning tree problem , 2002, Networks.

[20]  Fred W. Glover,et al.  Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions , 2002, Ann. Oper. Res..

[21]  Sudipto Guha,et al.  Approximation algorithms for directed Steiner problems , 1999, SODA '98.

[22]  Mark S. Daskin,et al.  Network and Discrete Location: Models, Algorithms and Applications , 1995 .

[23]  S. E. Dreyfus,et al.  The steiner problem in graphs , 1971, Networks.

[24]  Zvi Drezner,et al.  Optimal location with equitable loads , 2009, Ann. Oper. Res..

[25]  Bruno Simeone,et al.  Evaluation and Optimization of Electoral Systems , 1987 .

[26]  Martin W. P. Savelsbergh,et al.  Preprocessing and Probing Techniques for Mixed Integer Programming Problems , 1994, INFORMS J. Comput..

[27]  Polly Bart,et al.  Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph , 1968, Oper. Res..

[28]  Vipin Kumar,et al.  Graph partitioning for high-performance scientific simulations , 2003 .

[29]  Horst A. Eiselt,et al.  Decision analysis, location models, and scheduling problems , 2003 .

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