Organizing a social event - A difficult problem of combinatorial optimization

This interesting problem arose in the context of organizing a “progressive party” at a yachting rally. Some yachts were to be designated hosts; the crews of the remaining yachts would then visit the hosts for six successive half-hour periods. A guest crew could not revisit the same host, and two guest crews could not meet more than once during the evening. Additional constraints were imposed by the capacities of the host yachts and the crew sizes of the guests. Although this problem has many of the classical features of combinatorial optimization problems, it does not belong to any previously known class of problem. Integer programming formulations which included all the constraints resulted in very large models and all attempts to find a solution failed. However by solving a simple relaxation of the problem using linear programming we obtained a lower bound for the solution, which combined with a feasible solution obtained (spectacularly easily) by constraint programming led to an optimal solution. We describe our computational experience and discuss the features of this problem which account for the failure and success of the two approaches.