Int. J. Operational Research, Vol. 8, No. 4, 2010 399 Linear programming formulation of the set partitioning problem Moustapha Diaby Operations and Information Management, University of Connecticut, Storrs, CT 06268, USA E-mail: moustapha.diaby@business.uconn.edu Abstract: In this paper, we present a linear programming (LP) model of the set partitioning problem (SPP). The number of variables and the number of constraints of the proposed model are bounded by (third-degree) polynomial functions of the number of non-zero entries of the SPP input matrix, respectively. Hence, the model provides a new affirmative resolution to the all-important ‘P vs. NP’ question. We use a transportation problem-based reformulation that we develop, and a path-based modelling approach similar to that used in Diaby (2007) to formulate the proposed LP model. The approach is illustrated with a numerical example. Keywords: SPP; set partitioning problem; LP; linear programming; computational complexity; combinatorial optimisation. Reference to this paper should be made as follows: Diaby, M. (2010) ‘Linear programming formulation of the set partitioning problem’, Int. J. Operational Research, Vol. 8, No. 4, pp.399–427. Biographical notes: Moustapha Diaby is a Professor of Production and Operations Management at the University of Connecticut. He received a PhD in Management Science/Operations Research, MS in Industrial Engineering and BS in Chemical Engineering from the State University of New York at Buffalo. His teaching and research interests are in the areas of mathematical programming, manufacturing systems modelling and analysis, and supply chain and logistics management. His publications have appeared in top-tier journals such as European Journal of Operational Research, Int. J. Production Research, Journal of the Operational Research Society, Management Science, Operations Research, etc. He also serves/has served as a reviewer and/or ad hoc editorial team member for many of these journals. 1 Introduction One of the three most widely applied combinatorial optimisation problems (along with the travelling salesmen problem (see Samanlioglu et al., 2007) and the set covering problem (see Kinney et al., 2007)) is the set partitioning problem (SPP) (Balas and Padberg, 1976). The sheer volume of industrial applications of the problem that have been described in the literature precludes an exhaustive listing in a single journal paper. The richest source of applications in the recent literature – as well as the ‘older’ literature (see Balas and Copyright © 2010 Inderscience Enterprises Ltd.