On the Equal Sum Partition Problem

TL;DR

This paper investigates the equal sum partition problem, clarifies the boundary between solvable and unsolvable instances, and introduces new criteria and algorithms for specific cases.

Contribution

It extends the understanding of unsolvable instances satisfying the slack condition and develops a randomized algorithm for linear partitions with high probability of success.

Findings

Infinite families of unsolvable instances with specific ratios identified.

Slack condition is necessary and sufficient for the fractional relaxation.

Randomized rounding algorithm solves linear partitions with exponentially small failure probability.

Abstract

We consider the equal sum partition problem, motivated by distance magic graph labeling: Given $n,k \in \N$ such that $k\, | \sum_{i=1}^ni$ and a partition $p_1+\cdots+p_k=n$, when is it possible to find a partition of the set $\{1,2,\ldots,n\}$ into $k$ subsets of sizes $p_1,\dots,p_k$, such that the element sum in each subset is the same? A known necessary condition is the \emph{slack condition}, requiring that for all $j$, placing the largest possible elements in the $j$ smallest sets yields a total sum that is at least what is needed. However, this condition is not sufficient, and known counterexamples exist. This work clarifies the boundary between solvable and unsolvable instances of the problem. We extend the list of unsolvable problem instances satisfying the slack condition by exhibiting infinite families where the $n/k$ ratio is any rational number in the interval $(2,\frac{24}{7})$, and a new criterion for unsolvability. Furthermore, we show that the slack condition is natural, as it is both necessary and sufficient for the fractional relaxation of the problem. Based on this result, we prove that the problem is solvable for the class of linear partitions, where $k$ is fixed, $p_1,\ldots,p_k$ grow linearly with $n$, and where the slack condition holds in a strong sense. We do this by applying a randomized rounding algorithm to a solution of the fractional relaxation of the problem and proving that the algorithm has an exponentially small failure probability.