Computing minimal doubly resolving sets of graphs

• Published in: Computers & operations research Vol. 36; no. 7; pp. 2149 - 2159
• Main Authors: Kratica, Jozef, Čangalović, Mirjana, Kovačević-Vujčić, Vera
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01-07-2009; Elsevier; Pergamon Press Inc
• Subjects: Applied sciences; Dynamic programming; Evolutionary approach; Exact sciences and technology; Flows in networks. Combinatorial problems; Genetic algorithms; Graph coloring; Graph theory; Hypercubes; Integer programming; Linear programming; Mathematical programming; Metric dimension; Minimal doubly resolving sets; Operational research and scientific management; Operational research. Management science; Optimization techniques; Scheduling, sequencing; Studies; Hypercubes; Evolutionary approach; Metric dimension; Minimal doubly resolving sets; Genetic algorithms; Crew; Hamming distance; Enumeration; Evolutionary algorithm; Linear programming; Scheduling; Integer programming; Upper bound; Genetic algorithm; Hypercube; NP hard problem; Graph colouring; Timetabling problem; Metric; Large scale; Assembler; Dynamic programming
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2008.08.002

In this paper we consider the minimal doubly resolving sets problem (MDRSP) of graphs. We prove that the problem is NP-hard and give its integer linear programming formulation. The problem is solved by a genetic algorithm (GA) that uses binary encoding and standard genetic operators adapted to the problem. Experimental results include three sets of ORLIB test instances: crew scheduling, pseudo-boolean and graph coloring. GA is also tested on theoretically challenging large-scale instances of hypercubes and Hamming graphs. Optimality of GA solutions on smaller size instances has been verified by total enumeration. For several larger instances optimality follows from the existing theoretical results. The GA results for MDRSP of hypercubes are used by a dynamic programming approach to obtain upper bounds for the metric dimension of large hypercubes up to
2
90
nodes, that cannot be directly handled by the computer.