A Diversity-Aware Memetic Algorithm for the Linear Ordering Problem: Improving Best-Known Solutions for Standard Benchmarks

Abstract

The Linear Ordering Problem (LOP) is a very popular NP-hard combinatorial optimization problem with many practical applications that may require the use of large instances. The Linear Ordering Library (LOLIB) gathers a set of standard benchmarks widely used in the validation of solvers for the LOP. Among them, the xLOLIB2 collects some of the largest and most challenging instances in current literature. In this work, we present new best-known solutions for each of the 200 complex instances that comprises xLOLIB2. Moreover, the proposal devised in this research is able to achieve all current best-known solutions in the rest of instances of LOLIB and improve them in other 93 cases out of 485, meaning that important advances in terms of quality and robustness are attained. This important advance in the field of the LOP has been possible thanks to the development of a novel Memetic Algorithm (MA) that was designed by taking into account some of the weaknesses of state-of-the-art LOP solvers. One of the keys to success is that the novel proposal allows for a gradual shift from exploration to exploitation, which is done by taking into account the stopping criterion and elapsed period of execution to alter the internal decisions taken by the optimizer. The novel diversity-aware proposal is called the Memetic Algorithm with Explicit Diversity Management (MA-EDM) and extensive comparisons against state-of-the-art techniques provide insights into the reasons for the superiority of MA-EDM.