Two new integer linear programming formulations for the vertex bisection problem

Abstract

The vertex bisection problem (VBP) is an NP-hard combinatorial optimization problem with important practical applications in the context of network communications. The problem consists in finding a partition of the set of vertices of a generic undirected graph into two subsets (A and B) of approximately the same cardinality in such a way that the number of vertices in A with at least one adjacent vertex in B is minimized. In this article, we propose two new integer linear programming (ILP) formulations for VBP. Our first formulation (ILPVBP) is based on the redefinition of the objective function of VBP. The redefinition consists in computing the objective value from the vertices in B rather than from the vertices in A. As far as we are aware, this is the first time that this representation is used to solve VBP. The second formulation (MILP) reformulates ILPVBP in such a way that the number of variables and constraints is reduced. In order to assess the performance of our formulations, we conducted a computational experiment and compare the results with the best ILP formulation available in the literature (ILPLIT). The experimental results clearly indicate that our formulations outperform ILPLIT in (i) average objective value, (ii) average computing time and (iii) number of optimal solutions found. We statistically validate the results of the experiment through the well-known Wilcoxon rank sum test for a confidence level of 99.9%. Additionally, we provide 404 new optimal solutions and 73 new upper and lower bounds for 477 instances from 13 different groups of graphs.

Constructive heuristic for the vertex bisection problem

“…In the literature we can find both exact and approximate solution methods. Regarding the exact methods, seven approaches have been proposed for VBP, five Integer Linear Programming (ILP) formulations (Castillo-García & Hernández, 2019;Fraire et al, 2014;Jain, Saran & Srivastava, 2016a) and two branch and bound algorithms (Fraire et al, 2014;Jain, Saran, & Srivastava, 2016b). In addition, there are three metaheuristic algorithms (Herrán, Colmenar, & Duarte, 2019;Jain, Saran & 2016c;Terán-Villanueva et al, 2019) and one constructive algorithm for VBP (González et al, 2015), totalizing four approximate solutions.…”

“…LIT computes the objective value according to the traditional definition, that is, from the vertices in the set (see Figure 1). Conversely, CVBP uses the redefinition of the objective function proposed by Castillo-García and Hernández (2019). This means that CVBP computes the objective value from the vertices in the set .…”

“…This implies that CVBP is completely greedy. In the second experiment the best configuration of CVBP solved the entire benchmark of VBP consisting of 477 graphs of different classes and sizes (Castillo-García & Hernández, 2019). We also execute a random algorithm (RND) over the whole set of benchmark instances.…”