Comput Optim Appl (2013) 55:113–135 DOI 10.1007/s10589-012-9512-9 Mixed-Integer Linear Programming Formulations for the Software Clustering Problem Viviane Köhler · Marcia Fampa · Olinto Araújo Received: 8 February 2012 / Published online: 4 October 2012 © Springer Science+Business Media New York 2012 Abstract The clustering problem has an important application in software engineer- ing, which usually deals with large software systems with complex structures. To fa- cilitate the work of software maintainers, components of the system are divided into groups in such a way that the groups formed contain highly-interdependent modules and the independent modules are placed in different groups. The measure used to analyze the quality of the system partition is called Modularization Quality (MQ). Designers represent the software system as a graph where modules are represented by nodes and relationships between modules are represented by edges. This graph is referred in the literature as Module Dependency Graph (MDG). The Software Clus- tering Problem (SCP) consists in finding the partition of the MDG that maximizes the MQ. In this paper we present three new mathematical programming formulations for the SCP. Firstly, we formulate the SCP as a sum of linear fractional functions problem and then we apply two different linearization procedures to reformulate the problem as Mixed-Integer Linear Programming (MILP) problems. We discuss a preprocessing technique that reduces the size of the original problem and develop valid inequalities that have been shown to be very effective in tightening the formulations. We present numerical results that compare the formulations proposed and compare our results with the solutions obtained by the exhaustive algorithm supported by the freely avail- able Bunch clustering tool, for benchmark problems. V. Köhler · M. Fampa ( ) COPPE, Federal University of Rio de Janeiro, C.P. 68530, 21945-970, Rio de Janeiro, Brazil e-mail: fampa@cos.ufrj.br V. Köhler e-mail: vivianeck@gmail.com O. Araújo CTISM, Federal University of Santa Maria, Santa Maria, Rio Grande do Sul, Brazil e-mail: olinto@ctism.ufsm.br