Separating lifted odd-hole inequalitites to solve the index selection problem

The Index Selection Problem (ISP) is a phase of fundamental importance in the physical design of databases, calling for a set of indexes to be built in a database so as to minimize the overall execution time for a given database workload. The problem is a generalization of the well-known Uncapacitated Facility Location Problem (UFLP). In an earlier publication, we formulate ISP as a set packing problem, showing that our mathematical model contains all the clique inequalities, and describe a branch-and-cut algorithm based on the separation of odd-hole inequalities. In this paper, we describe an effective exact separation procedure for a suitably-defined family of lifted odd-hole inequalities, obtained by applying a Chvátal–Gomory derivation to the clique inequalities. Our analysis goes in the direction of determining a new class of inequalities over which efficient separation is possible, rather than introducing new classes of (facet-defining) inequalities that later turn out to be difficult to separate. Our separation procedure is embedded within our branch-and-cut algorithm for the exact solution of ISP. Computational results on two different classes of instances are given, showing the effectiveness of the new approach. We also test our algorithm on UFLP instances both taken from the literature and randomly generated.