Approximate extended formulations

Mixed integer programming (MIP) formulations are typically tightened through the use of a separation algorithm and the addition of violated cuts. Using extended formulations involving new variables is a possible alternative, but this often results in prohibitively large MIPs where even the linear programming relaxations are hard or impossible to solve. In this paper, we demonstrate how, in certain cases, it is possible and interesting to define "approximate" extended formulations. In all the examples considered, our description involves a single control parameter K. Large values of K result in strong but large formulations. In particular, when K takes its maximum value, the approximate formulation is identical to the complete extended formulation. Through this approximation parameter, the user has control over the tradeoff between the strength and the size of the formulation. Approximate extended formulations are proposed for a variety of lot-sizing problems and to the travelling salesman problem. We report computational results for several problems, including an industrial application and several small instances from the TSPLIB. The significant conclusion is that small values of the approximation parameter K are often sufficient to obtain excellent bounds.