Sperner theory
Algorithms for computing and integrating physical maps using unique probes
RECOMB '97 Proceedings of the first annual international conference on Computational molecular biology
File organization: the consecutive retrieval property
Communications of the ACM
PQ-tree algorithms.
Efficient text fingerprinting via Parikh mapping
Journal of Discrete Algorithms
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Journal of Computer and System Sciences
Consecutive ones property and PQ-trees for multisets: Hardness of counting their orderings
Information and Computation
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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. 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 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.