Improved Bounds for Poset Sorting in the Forbidden-Comparison Regime

We study the classical problem of sorting when comparison between certain pair of elements are forbidden. Along with the set of elements V, the input to our problem is an undirected graph G(V, E), whose edges represent the pairs that can be directly compared in constant time. We call this the comparison graph. It is also possible that the set of elements forms a partial-order, and not a total-order in which case, the sorting problem is the problem of determining all possible relations in the partial order, i.e. determining the (transitive) orientations of the edges of the graph.