No abstract

The Poset Cover Problem is an optimization problem where the goal is to determine a minimum set of posets that covers a given set of linear orders. This problem is relevant in the field of data mining, specifically in determining directed networks or models that explain the ordering of objects in a large sequential dataset. It is already known that the decision version of the problem is NP-Hard while its variation where the goal is to determine only a single poset that covers the input is in P. In this study, we investigate the variation, which we call the 2-Poset Cover Problem, where the goal is to determine two posets, if they exist, that cover the given linear orders. We derive properties on posets, which leads to an exact solution for the 2-Poset Cover Problem. Although the algorithm runs in exponential-time, it is still significantly faster than a brute-force solution. Moreover, we show that when the posets being considered are tree-posets, the running-time of the algorithm becomes polynomial, which proves that the more restricted variation, which we called the 2-Tree-Poset Cover Problem, is also in P.

No abstract

The Hammock(⏟𝟐𝟐, 𝟐𝟐 , … , 𝟐𝟐 𝒌𝒌 )-Poset Cover Problem is a variation of the Poset Cover Problem with the same input – set {𝑳𝑳𝟏𝟏, 𝑳𝑳𝟐𝟐, … , 𝑳𝑳𝒎𝒎} of linear orders over the set {𝟏𝟏, 𝟐𝟐, … ,𝒏𝒏}, but the solution is restricted to a set of simple hammock(𝟐𝟐⏟, 𝟐𝟐 , … , 𝟐𝟐 𝒌𝒌 ) posets. The problem is NP-Hard when 𝒌𝒌 ≥ 𝟑𝟑 but is in 𝑷𝑷 when 𝒌𝒌 = 𝟏𝟏. The computational complexity of the problem when 𝒌𝒌 = 𝟐𝟐 is not yet known. In this paper, we determine the approximation complexity of the cases that have been shown to be NP-Hard. We show that the Hammock(𝟐𝟐⏟, 𝟐𝟐 , … , 𝟐𝟐 𝒌𝒌 )-Poset Cover Problem is in 𝑨𝑨𝑨𝑨𝑨𝑨 and, in particular, (𝟏𝟏 + 𝟏𝟏 𝟐𝟐𝒌𝒌 )-approximable, for 𝒌𝒌 ≥ 𝟑𝟑. On the other hand, we also explore the computational complexity for the case where 𝒌𝒌 = 𝟐𝟐 [Hammock(2,2)-Poset Cover Problem]. We show that it is in 𝑷𝑷 when the transposition graph of the input set of linear orders is rectangular.