On complexity of optimized crossover for binary representations

TL;DR

This paper investigates the computational complexity of generating optimal offspring in crossover operations for binary representations, revealing polynomial solvability for some problems and NP-hardness for others.

Contribution

It establishes the complexity boundaries of the optimized gene transmitting crossover (OGTC) for Boolean linear programming problems, including polynomial solutions and NP-hardness results.

Findings

Polynomial solvability for maximum weight set packing and set partition problems

NP-hardness of certain OGTC cases in Boolean linear programming

Connection between OGTC and maximum weight independent set on hypergraphs

Abstract

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.