Counting the orderings for multisets in consecutive ones property and PQ-trees

Giovanni Battaglia, Roberto Grossi and Noemi Scutellà

Abstract. 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 QiR for some universe R, satisfies the C1P if the symbols in R can be permuted such that the elements of each set QiF occur consecutively, as a contiguous segment of the permutation of R’s 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-complete. We prove completeness results also for counting the permuta tions 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.

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