Consecutive Ones Property and PQ-Trees for Multisets: Hardness of Counting Their Orderings

A binary matrix satisfies the consecutive ones property (C1P) if its columns can be permuted such that the 1s in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q1, . . . , Qm}, where Qi ⊆ R for some universe R, satisfies the C1P if the symbols in R can be permuted such that the elements of each set Qi ∈ F occur consecutively, as a contiguous segment of the permutation of R’s symbols. We consider the C1P version on multisets and prove that counting its solutions is difficult (#P-complete). We prove completeness results also for counting the frontiers of PQ-trees, which are typically used for testing the C1P on sets, thus showing that a polynomial algorithm is unlikely to exist when dealing with multisets. We use a combinatorial approach based on parsimonious reductions from the Hamiltonian path problem, showing that the decisional version of our problems is therefore NP-complete.