Circulant graphs and GCD and LCM of subsets

Given two sets A and B of integers, we consider the problem of finding a set S⊆A of the smallest possible cardinality such the greatest common divisor of the elements of S∪B equals that of those of A∪B. The particular cases of B=∅ and #B=1 are of special interest and have some links with graph theor...

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Bibliographic Details
Published inInformation processing letters Vol. 115; no. 2; pp. 134 - 138
Main Authors von zur Gathen, Joachim, Shparlinski, Igor E.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 01.02.2015
Elsevier Sequoia S.A
Subjects
Analysis of algorithms
Approximation
Circulant graphs
Computational complexity
Computer science
Data processing
Graph algorithms
Graph theory
Graphs
Integer programming
Integers
Links
Mathematical analysis
Mathematical problems
Minimum cover
Studies
Graph algorithms
Analysis of algorithms
Circulant graphs
Minimum cover
Computational complexity
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ISSN0020-0190
1872-6119
DOI10.1016/j.ipl.2014.07.014

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Summary:Given two sets A and B of integers, we consider the problem of finding a set S⊆A of the smallest possible cardinality such the greatest common divisor of the elements of S∪B equals that of those of A∪B. The particular cases of B=∅ and #B=1 are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Set Cover Problem. •Finding minimal-size GCD preserving subsets of finite sets integers.•Finding minimal-size LCM preserving subsets of finite sets integers.•Reduction to the Set Cover Problem.
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ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2014.07.014