An integer linear programming formulation and genetic algorithm for the maximum set splitting problem

Abstract

We consider the maximum set splitting problem (MSSP). For the first time an integer linear programming (ILP) formulation is presented and validity of this formulation is given. We propose a genetic algorithm (GA) that uses the binary encoding and the standard genetic operators adapted to the problem. The overall performance of the GA implementation is improved by a caching technique. Experimental results are performed on two sets of instances from the literature: minimum hitting set and Steiner triple systems. The results show that CPLEX optimally solved all hitting set instances up to 500 elements and 10000 subsets. Also, it can be seen that GA routinely reached all optimal solutions up to 500 elements and 50000 subsets. The Steiner triple systems seems to be much more challenging for maximum set splitting problems since the CPLEX solved to optimality, within two hours, only two instances up to 15 elements and 35 subsets. For these instances GA reached all solutions as CPLEX but in much smaller running time.

A Variable Neighborhood Search Approach for Solving the Maximum Set Splitting Problem

“…The EM uses a fast local search procedure which additionally improves the efficiency of the overall system. The algorithm was examined on the same instances as in [14] and the results obtained clearly indicate that the proposed EM is a useful tool for solving MSSP.…”

“…Recent notable results in solving MSSP are presented in paper [14]. The authors introduced the first integer linear formulation (ILP) and proposed a genetic algorithm (GA) for solving MSSP.…”

“…In Section 4 the VNS algorithm for solving MSSP is described. Section 5 contains experimental results, based on the instances used in [14] and also in [13]. Section 6 is the conclusion.…”

“…Mathematical formulation. This section presents two already known integer programming formulations: quadratic integer formulation from [1] and integer linear programming (ILP) from [14].…”

A new mixed integer linear programming formulation for the maximum degree bounded connected subgraph problem

“…The first integer linear programming (ILP) formulation was proposed in [3]. There are great benefits for representing discrete optimization problems as integer linear programming models [14,3] or their generalization [20]. Since it has exponential number of constraints, its ability to solve MDBCSP instances is limited, so it is unable to solve even the instance presented in Example 1.…”