Upper Bounds on the Number of Solutions of Binary Integer Programs

We present a new method to compute upper bounds of the number of solutions of binary integer programming (BIP) problems. Given a BIP, we create a dynamic programming (DP) table for a redundant knapsack constraint which is obtained by surrogate relaxation. We then consider a Lagrangian relaxation of the original problem to obtain an initial weight bound on the knapsack. This bound is then refined through subgradient optimization. The latter provides a variety of Lagrange multipliers which allow us to filter infeasible edges in the DP table. The number of paths in the final table then provides an upper bound on the number of solutions. Numerical results show the effectiveness of our counting framework on automatic recording and market split problems.