Market split and basis reduction: Towards a solution of the Cornuejols–Dawande instances

Cornuejols and Dawande proposed a set of 0–1 linear programming instances that proved to be very hard to solve by traditional methods, and in particular by linear programming-based branch-and-bound. They offered these market split instances as a challenge to the integer programming community. The market split problem can be formulated as a system of linear diophantine equations in 0–1 variables. In our study we use the algorithm of Aardal, Hurkens, and Lenstra based on lattice basis reduction. This algorithm is not restricted to deal with market split instances only but is a general method for solving systems of linear diophantine equations with bounds on the variables. We show computational results from solving both feasibility and optimization versions of the market split instances with up to 7 equations and 60 variables and discuss various branching strategies and their effect on the number of enumerated nodes. To our knowledge, the largest feasibility and optimization instances solved before had 6 equations and 50 variables, and 4 equations and 30 variables, respectively. We also present a probabilistic analysis describing how to compute the probability of generating infeasible market split instances. By generating instances the way prescribed by Cornuejols and Dawande, one obtains relatively many feasible instances for sizes larger than 5 equations and 40 variables.