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.
翻译：跳跃是外形延伸中连续的元素,在原始外形中是无法比较的。 Arboreal跳跃数是一个硬硬的问题,目的是找到一个带有最小跳跃次数的给定外形的Arboreal扩展。本文的贡献有两个方面:(一)~a 特征特征,它揭示了Arboreal 扩展的跳跃次数与满足封面图某些结构特性的元素分隔大小之间的关系;(二)~~ 压缩整形编程模型和超常性来解决Arboreal跳跃数问题,同时比较两种战略的计算结果。精确方法为41个案例中的18个提供了最佳性证明,执行时间限于2小时。此外,我们的超常性能能够在3分钟内为所有情况找到可行的好解决办法。
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