BVNS for the Minimum Sitting Arrangement Problem in a Cycle

Signed graphs are a particular type of graph, in which the vertices are connected through edges labeled with a positive or negative weight. Given a signed graph, the Minimum Sitting Arrangement (MinSA) problem aims to minimize the total number of errors produced when the graph (named the input graph) is embedded into another graph (named the host graph). An error in this context appears every time that a vertex with two adjacent (one positive and one negative) is embedded in such a way that the adjacent vertex connected with a negative edge is closer than the adjacent vertex connected with a positive edge. The MinSA can be used to model a variety of real-world problems, such as links in online social networks, the location of facilities, or the relationships between a group of people. Previous studies of this problem have been mainly focused on host graphs, whose structure is a path. However, in this research, we compare two variants of the MinSA that differ in the graph used as the host graph (i.e., a path or a cycle). Particularly, we adapted a previous state-of-the-art algorithm for the problem based on Basic Variable Neighborhood Search. The solutions obtained are compared with the solutions provided by a novel Branch & Bound algorithm for small instances. We also analyze the differences found by using non-parametric statistical tests.