A formulation of the wide partition conjecture using the atom problem in discrete tomography

Abstract

The Wide Partition Conjecture (WPC) was introduced by Chow and Taylor as an attempt to prove inductively Rota's Basis Conjecture, and in the simplest case tries to characterize partitions whose Young diagram admits a "Latin" filling. Chow et al. (2003) showed how the WPC is related to problems such as edge-list coloring and multi-commodity flow. As far as we know, the conjecture remains widely open.

We show that the WPC can be formulated using the k-atom problem in Discrete Tomography, introduced in Gardner et al. (2000). In this approach, the WPC states that the sequences arising from partitions admit disjoint realizations if and only if any combination of them can be realized independently. This realizability condition can be checked in polynomial time, although is not sufficient in general Chen and Shastri (1989), Guinez et al. (2011). In fact, the problem is NP-hard for any fixed k >= 2 Durr et al. (2012). A stronger condition, called the saturation condition, was introduced in Guifiez et al. (2011) to solve instances where the realizability condition fails. We prove that in our case, the saturation condition is implied by the realizability condition. Moreover, we show that the saturation condition can be obtained as the Lagrangian dual of the linear programming relaxation of a natural integer programming formulation of the k-atom problem.