Computing minimal doubly resolving sets of graphs

Authors:
Jozef Kratica;Mirjana angalović;Vera Kovačević-Vujčić
Affiliations:
Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11000 Belgrade, Serbia;Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11000 Belgrade, Serbia;Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11000 Belgrade, Serbia
Venue:
Computers and Operations Research
Year:
2009

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Abstract

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 2^9^0 nodes, that cannot be directly handled by the computer.