The Sum Composition Problem

Abstract

In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\dots ,A_m]$ of $A$ where $m$ is the length of $B$ and the sum of each part $A_k$ is equal to the corresponding part of $B$. Then, we consider the following two problems: $i)$ the "exhaustive problem", consisting in the generation of all partitions of $A$ for which $B$ is sum composition of $A$, and $ii)$ the "existential problem", consisting in the verification of the existence of a partition of $A$ for which $B$ is sum composition of $A$. Starting from some general properties of the sum compositions, we present a first algorithm solving the exhaustive problem and then a second algorithm solving the existential problem. We also provide proofs of correctness and experimental analysis for assessing the quality of the proposed solutions along with a comparison with related works.