Realizing partitions respecting full and partial order information

For n ∈ , we consider the problem of partitioning the interval [0, n) into k subintervals of positive integer lengths `1, . . . , `k such that the lengths satisfy a set of simple constraints of the form `i ij `j where ij is one of <, >, or =. In the full information case, ij is given for all 1 ≤ i, j ≤ k. In the sequential information information case, ij is given for all 1 < i < k and j = i ± 1. That is, only the relations between the lengths of consecutive intervals are specified. The cyclic information case is an extension of the sequential information case in which the relationship 1k between `1 and `k is also given. We show that all three versions of the problem can be solved in time polynomial in k and log n.