# Vertex Bisection Problem

Herrán A., Colmenar J. M., Duarte A. (2018)

Optsicom project, University of Valencia (Spain)

## Problem Description

The Vertex Bisection Problem (VBP) belongs to the family of well-known graph partitioning problems, where the main goal is to find a partition of the vertices maximizing or minimizing a given objective function. In particular, VBP consists of dividing a graph into two parts *B* and *B’* with the same number of vertices, or differing in one if the number of vertices is odd, such as the vertex width (VW), defined as the number of vertices in *B* with at least one adjacent in *B’*, is minimized. As example, Figure (a) shows a connected undirected graph with 10 vertices and 13 edges. Figures (b) and (c) show two feasible solutions for the VBP with a vertex width of 2 and 3 respectively.

More formally, given a graph *G = (V,E)* where *V* is the set of vertices, *|V| = n*, and *E* is the set of edges, a feasible solution of the VBP is defined by two subsets of nodes *B* and *B’* (with *B* ∩ *B’* = ∅ and *B* ∪ *B’ = V*) where the size of each subset verifies that *B* = ⌊*n/2*⌋ and *B’* = *n* – ⌊*n/2*⌋, respectively. Hence, given a solution *S = (B, B’)*, its vertex width is computed as:

Let *𝒮* be the whole set of feasible partitions of *V* according to the aforementioned constraints. Therefore, the VBP consists of finding the solution *S* ∈ 𝒮* that minimizes the vertex width.

## State of the Art Methods

This problem has been dealt with from both, exact and heuristic perspectives. Among the exact methods, the most relevant works are:

**B&B**: Branch & Bound algorithm for the two proposed integer-linear programming models. Fraire (2014).**QCQP**: Quadratically constrained quadratic program for a different integer-linear programming model than the proposed in (Fraire, 2014). Jain (2016a).**B&B**: Branch & Bound algorithm that incorporates a greedy heuristic to produce an upper bound for the cost of the solutions. Jain (2016b).

The most relevant heuristic methods developed to solve this problem are:

**CH**: Different constructive heuristics. Castillo-Garcia (2015).**GA**: Genetic algorithm. Huacuja (2016).**MA**: Memetic algorithm. Pallavi (2016).

## Instances

Our experimental experience is developed on several sets of instances with different number of vertices, edges, and densities. These sets contain *Small-Full* (84), *Trees* (15), *Grids* (5) and *HB* (4) instances used in Fraire (2014), and the *Small* (23), *Harwell-Boeing* (36) and *Hypercubes* (8) instances used in (Pallavi, 2016).

We stored the resulting 175 instances as a library that we called VBP_LIB:

## Computational Experience

All the algorithms were implemented in C++ and the experiments were conducted on an Intel i7 6500U processor running at 2.5 GHz with 8 Gb of RAM using Windows 10. We have considered the VBP_LIB instances described above. The individual results for each instance can be downloaded here in Excel format.

Vertex Bisection problem results

## References

- H. Fraire, J. Terán-Villanueva, N.C. García, J. Gonzalez-Barbosa, E.R. del Angel, Y. Gómez-Rojas. Exact Methods for the Vertex Bisection Problem. Springer International Publishing (2014), pp. 567–577.
- P. Jain, G. Saran, K. Srivastava. A new integer linear programming and quadratically constrained quadratic programming formulation for vertex bisection minimization problem. J. Autom. Mobile Rob. Intell. Syst. 10(1) (2016a), pp. 69–73.
- P. Jain, G. Saran, K. Srivastava. Branch and bound algorithm for vertex bisection minimization problem. In: R.K. Choudhary, J.K. Mandal, N. Auluck, H.A. Nagarajaram (Eds.), Advanced Computing and Communication Technologies, Proceedings of the 9th ICACCT (2015), Springer Singapore, Singapore (2016b), pp. 17–23.
- N. Castillo-García, H.J.F. Huacuja, J.A.M. Flores, R.A.P. Rangel, J.J.G. Barbosa, J.M.C. Valadez. Comparative Study on Constructive Heuristics for the Vertex Separation Problem. Springer International Publishing, Cham (2015), pp. 465–474.
- H.J.F. Huacuja, N. Castillo-García. Optimization of the vertex separation problem with genetic algorithms. IGI Global (2016), pp. 13–31.
- J. Pallavi, G. Saran, K. Srivastava. On minimizing vertex bisection using a memetic algorithm. Inf. Sci. 369 (2016), pp. 765–787.