On the arboreal jump number of a poset

TL;DR

This paper investigates the NP-hard problem of finding an arboreal extension of a poset with the minimum number of jumps, providing a new characterization, an integer programming model, and a heuristic solution with computational results.

Contribution

It introduces a novel characterization linking jumps to partition structures, along with a compact integer programming model and a heuristic for solving the arboreal jump number problem.

Findings

Exact method solves 18 of 41 instances within two hours.

Heuristic finds good solutions for all instances in under three minutes.

The characterization relates jumps to the size of specific partitions of the poset.

Abstract

A jump is a pair of consecutive elements in an extension of a poset which are incomparable in the original poset. The arboreal jump number is an NP-hard problem that aims to find an arboreal extension of a given poset with minimum number of jumps. The contribution of this paper is twofold: (i)~a characterization that reveals a relation between the number of jumps of an arboreal order extension and the size of a partition of its elements that satisfy some structural properties of the covering graph; (ii)~a compact integer programming model and a heuristic to solve the arboreal jump number problem along with computational results comparing both strategies. The exact method provides an optimality certificate for 18 out of 41 instances with execution time limited to two hours. Furthermore, our heuristic was able to find good feasible solutions for all instances in less than three minutes.