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.