Some algorithmic results for [2]-sumset covers

Abstract : Let X={xi:1≤i≤n}⊂N+X={xi:1≤i≤n}⊂N+, and h∈N+h∈N+. The h-iterated sumset of X , denoted hX , is the set {x1+x2+...+xh:x1,x2,...,xh∈X}{x1+x2+...+xh:x1,x2,...,xh∈X}, and the [h][h]-sumset of X , denoted [h]X[h]X, is the set View the MathML source⋃i=1hiX. A [h][h]-sumset cover of S⊂N+S⊂N+ is a set X⊂N+X⊂N+ such that S⊆[h]XS⊆[h]X. In this paper, we focus on the case h=2h=2, and study the APX-hardproblem of computing a minimum cardinality [2]-sumset cover X of S (i.e. computing a minimum cardinality set X⊂N+X⊂N+ such that every element of S is either an element of X , or the sum of two - non-necessarily distinct - elements of X ). We propose two new algorithmic results. First, we give a fixed-parameter tractable (FPT) algorithm that decides the existence of a [2]-sumset cover of size at most k of a given set S . Our algorithm runs in View the MathML sourceO(2(3logk−1.4)kpoly(k)) time, and thus outperforms the O(5k2(k+3)2k2log(k)) time FPT result presented in Fagnot et al. (2009) [6]. Second, we show that deciding whether a set S has a smaller [2]-sumset cover than itself is NP-hard.
Document type :
Journal articles
Liste complète des métadonnées

Cited literature [18 references]  Display  Hide  Download
Contributor : Guillaume Fertin <>
Submitted on : Thursday, July 24, 2014 - 12:09:28 PM
Last modification on : Thursday, July 5, 2018 - 2:46:05 PM
Document(s) archivé(s) le : Tuesday, November 25, 2014 - 4:17:17 PM


Files produced by the author(s)


  • HAL Id : hal-01044891, version 1


Laurent Bulteau, Guillaume Fertin, Stéphane Vialette, Roméo Rizzi. Some algorithmic results for [2]-sumset covers. Information Processing Letters, Elsevier, 2015, 115 (1), pp.1-5. ⟨hal-01044891⟩



Record views


Files downloads