On the Facial Structure of the Alldifferent System

• Authors: D. Magos;I. Mourtos
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2011

We study the facial structure of the alldifferent system, i.e., the polytope (namely, $P_{I}$) defined as the convex hull of integer vectors satisfying such a system. We derive classes of facets for $P_{I}$ by examining induced subgraphs of the associated constraint graph. Some of these graphic structures (for example, odd holes, webs, etc.) are well known to induce facets of the set packing polytope, namely, $P_{S}$. This is a rather surprising result, given that $P_{S}$ is defined in terms of binary variables whereas $P_{I}$ is defined in terms of integer variables receiving values from a discrete domain. As a consequence, the resulting facet-defining inequalities are different for $P_{I}$ and $P_{S}$. Furthermore, we show that the families of facet-defining graphic structures for $P_{I}$ and $P_{S}$ do not coincide as we exhibit such a structure yielding facets for $P_{I}$ but not for $P_{S}$. We also prove that the facets of $P_{I}$ come in pairs of the form $(\alpha x\geq\beta,\alpha x\leq\delta)$ and show that the separation of the first implies that of the second (and vice versa). In addition, we provide the complete linear description of $P_{I}$ when the constraint graph of the alldifferent system is of degree 2.