Some algorithmic results for [2]-sumset covers

Abstract

Let X = {x i : 1 ≤ i ≤ n} ⊂ N + , and h ∈ N + . The h-iterated sumset of X, denoted hX, is the set {x 1 + x 2 + . . . + x h : x 1 , x 2 , . . . , x h ∈ X}, and theIn this paper, we focus on the case h = 2, and study the APX-hard problem of computing a minimum cardinality [2]-sumset cover X of S (i.e. computing a minimum cardinality set 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 . Second, we show that deciding whether a set S has a smaller [2]-sumset cover than itself is NP-hard.

Optimal monomial quadratization for ODE systems

“…Remark 2 (Relation to the [2]-sumset cover problem). The [2]-sumset cover problem [3] is, given a finite set S ⊂ Z >0 of positive integers, find a smallest set X ⊂ Z >0 such that S ⊂ X ∪ {x i + x j | x i , x j ∈ X}. It has been shown in [8, Proposition 1] that the [2]-sumset cover problem is APX-hard, moreover the set S used in the proof 1.…”