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arXiv · 1102.0041

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

Giovanni Battaglia·Roberto Grossi·Noemi Scutellà
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Abstract

A binary matrix satisfies the consecutive ones property (COP) if its columns can be permuted such that the ones in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q_1,..,Q_m}, where Q_i is subset of R for some universe R, satisfies the COP if the symbols in R can be permuted such that the elements of each set Q_i occur consecutively, as a contiguous segment of the permutation of R's symbols. We consider the COP 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 COP 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.

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Giovanni Battaglia, Roberto Grossi, Noemi Scutellà. 2011-01-31. Consecutive Ones Property and PQ-Trees for Multisets: Hardness of Counting Their Orderings. https://arxiv.org/abs/1102.0041

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