# On the heapability of finite partial orders

## TL;DR

This paper explores how to partition finite partial orders into the fewest heapable subsets, providing theoretical characterizations, flow-based algorithms, and greedy methods for special cases, along with open problems in the area.

## Contribution

It introduces a characterization similar to Dilworth's theorem for heapable decompositions and develops algorithms for minimal partitioning of partial orders.

## Key findings

- Characterization of heapable subset decompositions
- Flow-based algorithm for minimal decomposition
- Greedy algorithm for interval sets and sequences

## Abstract

We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing such a minimal decomposition. On the other hand, in the particular case of sets and sequences of intervals we prove that this minimal decomposition can be computed by a simple greedy-type algorithm. The paper ends with a couple of open problems related to the analog of the Ulam-Hammersley problem for decompositions of sets and sequences of random intervals into heapable sets.