title: NP-complete problems for systems of divisibilities of values of linear polynomials
metadata.name: dc.contributor.author
metadata.value: Kosovskii, Nikolay K.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Kosovskaya, Tatiana M.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Kosovskii, Nikolay N.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Starchak, Mikhail R.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2017-07-18T16:36:16Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2017-07-18T16:36:16Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2017-06
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Kosovskii N.K., Kosovskaya T.M., Kosovskii N.N., Starchak M.R. NP-complete problems for systems of divisibilities of values of linear polynomials. Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 2017, vol. 4 (62), issue 2, pp. 236–246.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu01.2017.207
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/6946
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: We study the algorithmic complexity of the subproblems of simultaneous divisibility of values of linear
  polynomials with non-negative coefficients in positive integers. In general, it is not known whether it is in
  NP, but it was previously proved it to be in NEXPTIME.
  The NP-completeness is proved for such two series of this problem, that in the first of them a number
  divides a linear polynomial, and in the second case linear polynomial is a divisor of a number. Values for
  the parameters were determined, for which the problem is NP-complete. For the values less than specified,
  it was proved the problem to be in the class P.
  We prove the NP-hardness of the particular case of the general problem of simultaneous divisibility
  of values of linear polynomials in which the coefficients of the variables may only take values from the set
  {1, 2} and the constant terms of linear polynomials may only take values from the set {1, 5}. The obtained
  results may be useful for determining the algorithmic complexity of the general problem.
  In Section 1, it is proved that the problem ∃x1 . . . xn &mj
  =1(Lj (x1 . . . xn) | K) is NP-complete for
  K ≥ 4 and exactly three non-zero coefficients in each linear polynomial. It is from the class P for 0 <
  K < 4.
  In Section 2, we studied the problem of simultaneous divisibility in positive integers from the nontrivial
  interval [D,D′], of values of linear polynomials with non-negative integer coefficients by a positive
  integer K: ∃x1 . . . xn ( &n i=1xi ∈ [D,D′] & &mj
  =1K | Lj (x1 . . . xn)). It is proved to be NP-complete for
  K ≥ 3. If the parameter of the problem is the number of non-zero coefficients in linear polynomials, the
  problem is NP-complete for only two non-zero coefficients. In the case of exactly one non-zero coefficient,
  the problem is from the class P.
  In Section 3, it is proved NP-hardness of ∃x1 . . . xn &mj
  =1(Lj (x1 . . . xn) | L′
  j (x1 . . . xn)), the problem
  of simultaneous divisibility of the values of linear polynomials on the values of the linear polynomials with
  the coefficients of the variables from {1, 2} and constant terms from {1, 5}. Refs 13.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 4 (62); Issue 2
metadata.lang: -
metadata.name: dc.subject
metadata.value: system of divisibilities of values of linear polynomials
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: NP-hardness
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: NP-completeness
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: NP-complete problems for systems of divisibilities of values of linear polynomials
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 07-Kosovsky.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/6946/1/07-Kosovsky.pdf
resources.format: Adobe PDF
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