Strong NP-Completeness of Unrestricted Balanced Mobiles

2026-09-21

AI Summary

本文解决了无限制平衡移动问题的强NP完全性，通过从数值3维匹配问题进行归约，并利用三个广泛分离的数值尺度来证明。

Abstract

A mobile is a rooted full binary tree whose leaves carry positive integer weights. The imbalance of an internal node is the absolute difference between the total weights of its two child subtrees, and the cost of the mobile is the sum of these imbalances. In the unrestricted \emph{Balanced Mobiles} problem, only the multiset of leaf weights is given: both the tree topology and the placement of the weights must be chosen so as to minimize the cost. The computational complexity of this unrestricted variant has remained open, although the variant with a prescribed topology is strongly NP-hard. We close this gap by proving that the decision version of unrestricted Balanced Mobiles is strongly NP-complete. Our reduction from Numerical 3-Dimensional Matching with Distinct Integers uses three widely separated numerical scales. Tight telescoping bounds force every threshold-achieving mobile into a canonical hierarchy, after which pairwise distinctness of the source integers collapses the hierarchy to single triples from which a valid numerical matching can be recovered.

Problem

• unrestricted Balanced Mobiles
• strong NP-completeness
• numerical 3-dimensional matching

Innovation

• Strong NP-Completeness
• Unrestricted Balanced Mobiles
• Numerical 3-Dimensional Matching
• Canonical Hierarchy