The 3-Partition problem (wiki) is a $\text{NP}$-complete problem which is to decide whether a given multiset of integers can be partitioned into triples that all have the same sum. It is well-known that the 3SAT problem has a plenty of variants. Are there some variants of the 3-Partition problem discussed in the literature?

  • $\begingroup$ Did you try searching for any? You could immediately consider a variant where each integer occurs exactly or at most $k$ times, for some fixed $k$. $\endgroup$ – Juho Mar 8 '14 at 13:08
  • $\begingroup$ @Juho I searched and found little. Thanks for your hint. I would prefer to the variants which have been considered in the literature. $\endgroup$ – hengxin Mar 8 '14 at 13:13

One variant of 3-partition is the following distinct 3-partition problem, shown to be NP-complete in [1].

Distinct 3-partition

Input: A set $X = \{a_1,a_2,\ldots,a_{3n}\}$ of positive distinct integers, and a positive integer $B$ where $\Sigma_{i=1}^{3n} a_i = nB$, and $B/4 < a_i < B/2$, where $i \in [3n]$.

Question: Is there a partition of $X$ into $n$ triples such that the elements in each triple sum to $B$?

The problem is also NP-complete when $B$ is bounded above by a polynomial in $n$. For an "application", a reduction from the problem is used in [2] to show a certain graph problem is hard.

[1] H. Hulett, T.G. Will, G.J. Woeginger, Multigraph realizations of degree sequences: Maximization is easy, minimization is hard, Operations Research Letters 36 (2008) 594–596.

[2] Bonato, Anthony, Jeannette Janssen, and Elham Roshanbin. "Burning a graph is hard." arXiv preprint arXiv:1511.06774 (2015).


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