5. Concluding Remarks 6. Acknowledgement

23] M. Raghavachari. On connections between zero-one integer programming and concave programming under linear constraints. 26 5] T.A. Feo and M.G.C. Resende. A probabilistic heuristic for a computationally diicult set covering problem. Two computationally diicult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. 16] K. Levenberg. A method for the solution of certain problems in least squares. Quart. In this paper, we have introduced an interior point algorithm for zero-one integer programming feasibility. We have described in detail the procedure of generating a descent direction for a potential function whose global minimum corresponds to a feasible zero-one integer solution. We show how to incorporate this descent direction procedure in a general purpose algorithm for zero-one integer programming. To illustrate the applicability of this general purpose algorithm, we use it to solve several instances of computationally diicult set covering problems that arise from computing the 1-width of the incidence matrix of Steiner triple systems. In that context, we show how to start the algorithm, propose two rounding schemes and show how to proceed when a nonglobal local minimum is encountered. We have found optimal covers for two instances with known optimal solutions and the best known covers for instances varying in size from 81 variables and 1080 constraints to 243 variables and 9801 constraints. The code used to produce the experimental results solves the descent direction linear system by direct Cholesky factorization, taking no advantage of problem structure. One may expect improved performance with an implementation under development that takes advantage of sparsity and nonzero structure of the linear system. References 1] D. Avis. A note on some computationally diicult set covering problems. Mathematical Programming, 18:138{145, 1980. 2] D. O. Bausch. Computational advances in the solution of large-scale set covering and set partitioning problems. Master's thesis, A note on zero-one integer and concave programming.