## Abstract

The tuple of permutations and the tuple of indices are required to be found in the problem considered in order to minimize the sum of elements chosen from the given family of finite numerical sequences subject to some constraints on the elements choice. Namely, given the family of L numerical nonnegative N-element sequences and a positive integer J, it is required to minimize the sum of J intra-sums. Each element corresponds to one element in one of L input sequences, and all possible L-permutations are admissible in this one-to-one correspondence in each intra-sum of L elements. In addition, there are some constraints on the indices of the summed sequence elements. The problem solution is a pair of tuples, namely, (1) a tuple of J permutations on L elements, and (2) a tuple of JL increasing indices. The paper presents an exact polynomial-time algorithm with O(N
^{5}
) running time for this problem. In particular, the problem is induced by an applied problem of noiseproof searching for repetitions of the given tuple of elements with their possible permutations at each tuple repeat, and finding the positions of these elements in the numerical sequence distorted by noise under some constraints on unknown positions of elements. The applied problem noted is related, for example, to the remote monitoring of several moving objects with possible arbitrary displacements (permutations) of these objects.

Original language | English |
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Title of host publication | Analysis of Images, Social Networks and Texts - 7th International Conference, AIST 2018, Revised Selected Papers |

Editors | Alexander Panchenko, Wil M. van der Aalst, Michael Khachay, Panos M. Pardalos, Vladimir Batagelj, Natalia Loukachevitch, Goran Glavaš, Dmitry I. Ignatov, Sergei O. Kuznetsov, Olessia Koltsova, Irina A. Lomazova, Andrey V. Savchenko, Amedeo Napoli, Marcello Pelillo |

Publisher | Springer-Verlag GmbH and Co. KG |

Pages | 305-317 |

Number of pages | 13 |

ISBN (Print) | 9783030110260 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

Event | 7th International Conference on Analysis of Images, Social Networks and Texts, AIST 2018 - Moscow, Russian Federation Duration: 5 Jul 2018 → 7 Jul 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11179 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 7th International Conference on Analysis of Images, Social Networks and Texts, AIST 2018 |
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Country | Russian Federation |

City | Moscow |

Period | 05.07.2018 → 07.07.2018 |

## Keywords

- Exact polynomial-time algorithm
- Finite numerical sequences
- Optimal summing
- Permutations