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**Common intervals and permutation reconstruction from MinMax-betweenness constraints.**
*(English)*
Zbl 1400.68150

Summary: The MinMax-Betweenness problem is defined as follows. We are given a positive integer \(n\) and, for each \(t = 0, 1, 2, \dots, n\), two integers \(m_t\) and \(M_t\) with \(m_t \leq t\) and \(t + 1 \leq M_t\) (this is called a MinMax-profile). The question is: Is there a permutation \(P\) on \(\{0, 1, 2, \dots, n + 1 \}\) such that \(m_t\) is the minimum and \(M_t\) is the maximum element of \(P\) located between \(t\) (included) and \(t + 1\) (included), assuming 0 is the leftmost and \(n + 1\) is the rightmost element of \(P\)?

We consider here the directed variant of the problem, where the left-to-right order of \(t\) and \(t + 1\) on \(P\) is known for each \(t = 0, 1, 2, \dots, n\). Whereas the complexity of the general directed or undirected problem is open, the particular case of the directed variant where the intervals \([m_t . . M_t]\) (\(t \neq 0, n + 1\)), containing the integers between \(m_t\) (included) and \(M_t\) (included), are linearly ordered by inclusion is polynomially solvable. In this case, the MinMax-profile is called linear.

In this paper, we use separable MinMax-subprofiles, that are intimately related to common intervals, to deal with MinMax-profiles – that we name L-reducible – which are not linear, but present decomposition properties allowing us to handle them using linear MinMax-(sub)profiles. We show that for L-reducible MinMax-profiles the Directed MinMax-Betweenness problem is solvable in polynomial time. We also give a polynomial algorithm to recognize L-reducible MinMax-profiles, with running time of \(O(n^2)\).

Moreover, we show that the DirectedMin-Betweenness (resp. DirectedMax-Betweenness) problem, where only \(m_t\) (resp. only \(M_t\)) is given for each \(t = 0, 1, 2, \dots, n\), is polynomial.

We consider here the directed variant of the problem, where the left-to-right order of \(t\) and \(t + 1\) on \(P\) is known for each \(t = 0, 1, 2, \dots, n\). Whereas the complexity of the general directed or undirected problem is open, the particular case of the directed variant where the intervals \([m_t . . M_t]\) (\(t \neq 0, n + 1\)), containing the integers between \(m_t\) (included) and \(M_t\) (included), are linearly ordered by inclusion is polynomially solvable. In this case, the MinMax-profile is called linear.

In this paper, we use separable MinMax-subprofiles, that are intimately related to common intervals, to deal with MinMax-profiles – that we name L-reducible – which are not linear, but present decomposition properties allowing us to handle them using linear MinMax-(sub)profiles. We show that for L-reducible MinMax-profiles the Directed MinMax-Betweenness problem is solvable in polynomial time. We also give a polynomial algorithm to recognize L-reducible MinMax-profiles, with running time of \(O(n^2)\).

Moreover, we show that the DirectedMin-Betweenness (resp. DirectedMax-Betweenness) problem, where only \(m_t\) (resp. only \(M_t\)) is given for each \(t = 0, 1, 2, \dots, n\), is polynomial.

### MSC:

68R05 | Combinatorics in computer science |

05A05 | Permutations, words, matrices |

68Q25 | Analysis of algorithms and problem complexity |

68W05 | Nonnumerical algorithms |

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\textit{L. Pelletier} and \textit{I. Rusu}, J. Discrete Algorithms 49, 8--26 (2018; Zbl 1400.68150)

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### References:

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