A hybrid of max-min ant system and linear programming for the k-covering problem

This paper presents a hybrid algorithm of linear programming (LP), max-min ant system, and local search for solving large instances of the k-covering problem (SCkP). This algorithm exploits the LP-relaxation solution by classifying the columns, based on their reduced costs, into three sets, such that two of these sets have the columns that need to be included or excluded from any solution while ants search the third set, the selection set, to construct their feasible solutions. Moreover, to choose high-quality columns from the selection set, ants rely on heuristic information derived from the rows' dual costs, which we obtain from the LP-relaxation solution as well. To benchmark our algorithm, we solve a set of 135 instances and compare the results with those of the state-of-the-art algorithm, in addition to the best-known solutions obtained using a branch and bound algorithm. Our algorithm shows superior results in terms of solution quality and computation time. Moreover, it can identify two new best-known solutions. HighlightsUsed linear programming relaxation to reduce the search region, to have a partial starting solution, and to develop an effective heuristic that ants can use to select columns when generating solutions.Achieved a considerable reduction in computation time compared to previous work by reducing the search region and not applying the local search algorithm to all solutions.To construct a solution, an ant first chooses a row to cover based on a heuristic then it chooses a column to cover. The heuristic used to choose rows chooses the row that needs the least number of columns to satisfy the cover constraints.Tried a new method to control the pheromone limits and mathematically showed that it did not have much contribution to the good results presented in this paper.Suggested a new search region reduction methodology that is suitable to all covering problems.