The Wide Partition Conjecture and the Atom Problem in Discrete Tomography

The Wide Partition Conjecture (WPC) was introduced by Chow and B. 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. T. Chow, C. K. Fan, M. Goemans, and J. Vondrak. Wide partitions, latin tableaux, and rota?s basis conjecture. Adv. in Appl. Math., 31(2):334–358, 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 C. Dürr, F. Guíñez, and M. Matamala. Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography. SIAM J Discrete Math, 26(1):330, 2012.]. 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 realizable independently. This realizability condition is not sufficient in general. A stronger condition, the saturation condition, was used in F. Guíñez, M. Matamala, and S. Thomassé. Realizing disjoint degree sequences of span at most two: A tractable discrete tomography problem. Discrete Appl.Math., 159(1):23–30, 2011] to solve instances were the realizability condition fails. We prove that in our case, the saturation condition is satisfied providing the realizability condition does. Moreover, we show that the saturation condition can be obtained from the Langrangean dual of a natural LP formulation of the k-atom problem.