Realizing partitions respecting full and partial order information

For n∈Nn∈N, we consider the problem of partitioning the interval [0,n)[0,n) into k   subintervals of positive integer lengths ℓ1,…,ℓkℓ1,…,ℓk such that the lengths satisfy a set of simple constraints of the form ℓi⋄ijℓjℓi⋄ijℓj where ⋄ij⋄ij is one of <, >, or =. In the full information   case, ⋄ij⋄ij is given for all 1⩽i,j⩽k1⩽i,j⩽k. In the sequential information   case, ⋄ij⋄ij is given for all 1