On Complexity of Optimized Crossover for Binary Representations

We consider the computational complexity of producing the best possible offspring in a crossover, given two solutions of the parents. The crossover operators are studied on the class of Boolean linear programming problems, where the Boolean vector of variables is used as the solution representation. By means of efficient reductions of the optimized gene transmitting crossover problems (OGTC) we show the polynomial solvability of the OGTC for the maximum weight set packing problem, the minimum weight set partition problem and for one of the versions of the simple plant location problem. We study a connection between the OGTC for linear Boolean programming problem and the maximum weight independent set problem on 2-colorable hypergraph and prove the NP-hardness of several special cases of the OGTC problem in Boolean linear programming.