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}F={Q1,…,Qm}, where Qi⊆RQi⊆R for some universe R, satisfies the c1p if the symbols in R   can be permuted such that the elements of each set Qi∈FQi∈F occur consecutively, as a contiguous segment of the permutation of Rʼs symbols. Motivated by combinatorial problems on sequences with repeated symbols, we consider the c1p version on multisets   and prove that counting the orderings (permutations) thus generated is #P#P-complete. We prove completeness results also for counting the permutations generated by PQ-trees (which are related to the c1p), thus showing that a polynomial-time algorithm is unlikely to exist when dealing with multisets and sequences with repeated symbols. To prove our results, we use a combinatorial approach based on parsimonious reductions from the Hamiltonian path problem, which enables us to prove also the hardness of approximation for these counting problems.