Sparsity of integer formulations for binary programs

This paper considers integer formulations of binary sets X of minimum sparsity, i.e., the maximal number of non-zeros for each row of the corresponding constraint matrix is minimized. Providing a constructive mechanism for computing the minimum sparsity, we derive sparsest integer formulations of several combinatorial problems, including the traveling salesman problem. We also show that sparsest formulations are NP-hard to separate, while (under mild assumptions) there exists a dense formulation of X separable in polynomial time.