无限制平衡移动树的强NP完全性

中文摘要 AI 辅助

移动树是一棵有根满二叉树，其叶子携带正整数权重。内部节点的失衡度是其两个子树总权重之差的绝对值，移动树的代价是所有这些失衡度之和。在无限制的“平衡移动树”问题中，仅给定叶子权重的多重集：必须同时选择树拓扑和权重放置方式以最小化代价。尽管具有指定拓扑的变体是强NP难的，但此无限制变体的计算复杂性一直悬而未决。我们通过证明无限制平衡移动树的判定版本是强NP完全的来填补这一空白。我们的归约来自具有不同整数的数值三维匹配问题，使用了三个广泛分离的数值尺度。紧致的望远镜式界限迫使每个达到阈值的移动树进入一个规范层次结构，之后源整数的两两不同性将该层次结构折叠为单个三元组，从中可以恢复有效的数值匹配。

英文摘要

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.

发表机构

• Faculty of Mathematics and Computer Science, University of Bucharest（布加勒斯特大学数学与计算机科学系）

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