Journey to the center of the linear ordering problem

A number of local search based algorithms have been designed to escape from the local optima, such as, iterated local search or variable neighborhood search. The neighborhood chosen for the local search as well as the escape technique play a key role in the performance of these algorithms. Of course, a specific strategy has a different effect on distinct problems or instances. In this paper, we focus on a permutation-based combinatorial optimization problem: the linear ordering problem. We provide a theoretical landscape analysis for the adjacent swap, the swap and the insert neighborhoods. By making connections to other different problems found in the Combinatorics field, we prove that there are some moves in the local optima that will necessarily return a worse or equal solution. The number of these non-better solutions that could be avoided by the escape techniques is considerably large with respect to the number of neighbors. This is a valuable information that can be included in any of those algorithms designed to escape from the local optima, increasing their efficiency.TIN2016-78365-R IT1244-1

Characterization of rankings generated by pseudo-Boolean functions

In this paper we pursue the study of pseudo-Boolean functions as ranking generators. The objective of the work is to find new insights between the relation of the degree of a pseudo-Boolean function and the rankings that can be generated by these insights. Based on a characterization theorem for pseudo-Boolean functions of degree , several observations are made. First, we verify that pseudo-Boolean functions of degree , where is the search space dimension, cannot generate all the possible rankings of the solutions. Secondly, the sufficient condition for a ranking to be generated by a pseudo-Boolean function of dimension is presented, and also the necessary condition is conjectured. Finally, we observe that the same argument is not sufficient to prove which ranking can be generated by pseudo-Boolean functions of degree .This research has been partially supported by the Spanish Ministry of Science and Innovation [PID2022-137442NB-I00, PID2019-104933GB-I00 funded by MCIN/AEI/10.13039/501100011033] and BCAM Severo Ochoa accreditation, Spain [CEX2021-001142-S]; the Basque Government, Spain [BERC 2022–2025. IT1504-22, IT1494-22]; and the University of the Basque Country UPV/EHU [GIU20/054]. Open Access funding provided by University of Basque Country