Block-insertion-based algorithms for the linear ordering problem

The linear ordering problem (LOP) is an NP-hard combinatorial optimization problem with wide applications. As the problem is computationally intractable, the only practical alternative is to develop efficient heuristics to solve it. Here we present four new properties of block insertion for the LOP, and show that changing the node order in one sub-problem does not affect the objective values of the remaining sub-problems. Based on these properties, we then propose three local search schemes for solving the LOP. Our experimental results show that the block insert with the first strategy often outperforms other local search schemes within the same computational time. To further improve the performance of this local search scheme, we incorporate it into the iterated local search and genetic algorithm frameworks, and develop the block-insertion-based iterated local search (ILS_b) and memetic algorithm (MA_b), respectively. The computational results show that both the ILS_b and MA_b outperform the state-of-the-art meta-heuristics. Moreover, with appropriate parameter settings, the MA_b frequently outperforms the ILS_b. Finally, we design a parallel computing framework, which divides the LOP problem into independent sub-problems that are solved in parallel by exact methods. This parallel framework can further improve the solutions derived by MA_b or other heuristics.