# On a Problem of Summing Elements Chosen from the Family of Finite Numerical Sequences

## 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 \(\mathcal {O}(N^5)\) 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.

## Keywords

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

### Acknowledgments

The study presented in Sect. 3 was supported by the Russian Science Foundation, project 16-11-10041. The study presented in Sects. 2 and 4 was supported by the Russian Foundation for Basic Research, projects 16-07-00168, and by the Russian Academy of Science (the Program of Basic Research), project 0314-2016-0015, and by the Russian Ministry of Science and Education under the 5–100 Excellence Programme.

## References

- 1.Kel’manov, A.V., Mikhailova, L.V., Khamidullin, S.A.: A posteriori joint detection of a recurring tuple of reference fragments in a quasi-periodic sequence. Comput. Math. Math. Phys.
**48**(12), 2276–2288 (2008)MathSciNetCrossRefGoogle Scholar - 2.Papadimitrou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Upper Saddle River (1982)Google Scholar
- 3.Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar