Sparsity of integer formulations for binary programs

Christopher Hojny, Hendrik Lüthen, Marc E. Pfetsch

Abstract: This paper considers integer formulations of binary sets X<math><mi is="true">X</mi></math> 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<math><mi mathvariant="script" is="true">NP</mi></math>-hard to separate, while (under mild assumptions) there exists a dense formulation of X<math><mi is="true">X</mi></math> separable in polynomial time.