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English version:
Journal of Applied and Industrial Mathematics, 2017, 11:3, 334-346

Volume 24, No 3, 2017, P. 104-124

UDC 519.6
V. M. Fomichev
Computational complexity of the original and extended Diophantine Frobenius problem

Abstract:
We deduce the law of nonstationary recursion which makes it possible, for given a primitive set $A = {a_1, . . . , a_k}$, $k > 2$, to construct an algorithm for finding the set of the numbers outside the additive semigroup generated by $A$. In particular, we obtain a new algorithm for determining the Frobenius numbers $g(a_1, . . . , a_k)$. The computational complexity of these algorithms is estimated in terms of bit operations. We propose a two-stage reduction of the original primitive set to an equivalent primitive set that enables us to improve complexity estimates in the cases when the two-stage reduction leads to a substantial reduction of the order of the initial set.
Bibliogr. 16.

Keywords: Frobenius number, primitive set, additive semigroup, computational complexity.

DOI: 10.17377/daio.2017.24.537

1. Financial University under the Government of the Russian Federation,
49 Leningradsky Ave., 125993 Moscow, Russia,
2. National Research Nuclear University MEPhI,
31 Kashirskoe Highway, 115409 Moscow, Russia
e-mail: fomichev@nm.ru

Revised 31 October 2016

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