Approximation and Computational Complexity of Some Hammock Variations of the Poset Cover Problem.

The Hammock(2, 2 ,..., 2/k)-Poset Cover Problem is a variation of the Poset Cover Problem with the same input - set {L1, L2, ..., Lm} of linear orders over the set {1, 2, ..., n}, but the solution is restricted to a set of simple hammock(2, 2 ,..., 2/k) posets. The problem is NP-Hard when k = 3 but is in P when k = 1. The computational complexity of the problem when k = 2 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(2, 2 ,..., 2/k)-Poset Cover Problem is in APX and, in particular, (1 + 1/2K)-approximable, for k = 3. On the other hand, we also explore the computational complexity for the case where k = 2 [Hammock(2,2)-Poset Cover Problem]. We show that it is in P when the transposition graph of the input set of linear orders is rectangular. [ABSTRACT FROM AUTHOR]