Heuristic methods to consecutive block minimization

• Published in: Computers & operations research 2020-08, Vol.120, p.104948-11, Article 104948
• Main Authors: Soares, Leonardo C.R., Reinsma, Jordi Alves, Nascimento, Luis H.L., Carvalho, Marco A.M.
• Format: Article
• Language: English
• Subjects: Algorithms; Benchmarks; Combinatorial analysis; Combinatorial problems; Consecutive block minimization; Exact algorithms; Graph representations; Graph theory; Graphical representations; Heuristic methods; Iterated local search; Metaheuristic; Operations research; Optimization; Permutations; Traveling salesman problem
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2020.104948

Many combinatorial problems addressed in the literature are modeled using binary matrices. It is often of interest to verify whether these matrices hold the consecutive ones property (C1P), which implies that there exists a permutation of the columns of the matrix such that all nonzero elements can be placed contiguously, forming a unique 1-block in every row. The minimization of the number of 1-blocks is approached by a well-known problem in the literature called consecutive block minimization (CBM), an NP-hard problem. In this study, we propose a graph representation, a heuristic based on a classical algorithm in graph theory, the implementation of a metaheuristic for solving the CBM and the application of an exact method based on a reduction of the CBM to a particular version of the well-known traveling salesman problem. Computational experiments demonstrate that the proposed metaheuristic implementation is competitive, as it matches or improves the best known solution values for all benchmark instances available in the literature, except for a single instance. The proposed exact method reports, for the first time, optimal solutions for these benchmark instances. Consequently, the proposed methods outperform previous methods and become the new state-of-the-art for solving the CBM.