A genetic algorithm for the minimum generating set problem

Abstract

Given a set of positive integers S, the minimum generating set problem consists in finding a set of positive integers T with a minimum cardinality such that every element of S can be expressed as the sum of a subset of elements in T. It constitutes a natural problem in combinatorial number theory and is related to some real-world problems, such as planning radiation therapies.We present a new formulation to this problem (based on the terminology for the multiple knapsack problem) that is used to design an evolutionary approach whose performance is driven by three search strategies; a novel random greedy heuristic scheme that is employed to construct initial solutions, a specialized crossover operator inspired by real-parameter crossovers and a restart mechanism that is incorporated to avoid premature convergence. Computational results for problem instances involving up to 100,000 elements show that our innovative genetic algorithm is a very attractive alternative to the existing approaches.