The Maximum Zero-Sum Partition problem

Abstract

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset $S=\{a_1,a_2,\dots ,a_n\}$ of integers $a_i\in Z^{⁎}$ (where $Z^{⁎}$ denotes the set of non-zero integers) such that ${\sum }_{i=1}^n{a}_i=0$, find a maximum cardinality partition $\{S_1,S_2,\dots ,S_k\}$ of $S$ such that, for every $1\le i\le k$, ${\sum }_{a_j\in S_i}{a}_j=0$. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in $S$ and (iii) the largest integer in $S$.