Feasible Encodings for GA Solutions of constrained Minimal Spanning Tree Problems

We solve by a genetic algorithm (GA), three NPhard, constrained, minimal spanning tree (MST) problems on a complete graph using a novel encoding for the genotype which ensures feasibility of the search space when performing crossover and mutation and when initializing the population. By employing a feasible encoding the standard, mainstream, GA paradigm is preserved allowing us to capture the full benefits of the genetic operators. We do not need to resort to the elaborate and time consuming detection and repair operations proposed by other researchers to handle non feasible population members. The feasible genotype for the degree constrained or leaf constrained MST problem is a permutation code which is suitably mapped into a Prufer code (phenotype) whose spanning tree obeys the desired tree constraints. This methodology can provide a framework for the development of feasible GA encodings for a wide class of other constrained minimal spanning tree problems on a complete graph. The effectiveness and competitive performance of this methodology is demonstrated by convergence to a global optimum for various degree constrained MST problems including a benchmark problem often cited in the literature. We investigate the Genetic Algorithm ( GA ) solutionsof three NP-hard constrained MST problems on aweighted complete graph, viz, the order constrained MSTproblem with time dependent edge weights, the degreeconstrained minimal spanning tree (DCMST) problemand the leaf constrained minimal spanning tree (LCMST)problem . These important problems are of practical usein circuit and communications network design.The unique strength of the paper is the development ofinnovative encodings for the genotype of the populationmembers which ensures feasibility of the search spacewhen performing the genetic operations of crossover andmutation and when generating the initial population .By employing a feasible encoding, the standard,mainstream, GA paradigm is preserved allowing us tocapture the full benefits of the genetic operators and touse , in a straightforward manner, any of the numerousenhancement techniques in the GA literature. We do notneed to resort to the elaborate and time consumingdetection and repair operations proposed by otherresearchers to handle non feasible population members.Our method avoids the loss of important genetic materialinherent in repair mechanisms.Building on our prior work with Prufer codes for feasibleencodings of spanning trees [Edelson and Gargano,1998] , we develop novel feasible GA encodings for theDCMST and LCMST problems. The genotype for eachproblem is a suitable permutation code (an adaptation ofthe simple idea of a permutation) which is feasible sincepermutation codes always give rise to feasible searchspaces. Appropriate linear mappings are developed whichtransform the permutation code into a Prufer code(phenotype) whose spanning tree obeys the desired treeconstraints. This methodology can provide aframework for the development of feasible GA encodingsfor a wide class of other constrained minimal spanningtree problems on a complete graph.The GA methodology insures diversity of the populationin that all Prufer codes (and their correspondingconstrained spanning trees) are equally likely to be in theinitial population and be selected as one of the parentpairs during mating. We demonstrate the effectivenessand competitive performance of this method by using acustomized Visual C++ program to generatecomputational results for four DCMST problemsincluding a benchmark problem often cited in theliterature. Convergence to the global optimum isachievable with high mutation rates. ReferencesEdelson,W. , Gargano, M.L., Minimal edge-OrderedSpanning Trees Solved by a Genetic Algorithm WithFeasible Search Space, Congressus Numerantium 135,(1998), 37-45.