图的高效重组，第2部分：平衡情况
Efficient Reassembling of Graphs, Part 2: The Balanced Case
摘要：
简单连通图G =（V，E）的重组是对早期网络分析研究中出现的一个问题的抽象。重组过程有一个简单的公式（有几个等价的公式）相对于二叉树B（重组树），其中根结点在顶部并且$n$叶节点在底部，其中每个横截面对应于V的分区，使得：- 底部（或第一）横截面（所有的叶子）是V的具有n个单顶点块的最好划分，- 顶部（或最后）横截面（根）是具有单个块的粗划分，整个集合V，- 节点（或块）在中间横截面中（或分区）是合并其两个子节点的结果（或块）在横截面中在重组过程中遇到的最大边边界度就是我们所说的重组的阿尔法测度，所有边边界度的和就是它的β测度。阿尔法优化（Alpha Optimization）β-优化）是确定最小化其α-测度（分别为β测量）。有不同形式的重组。在早期的报告中，我们研究了线性重组，这是当B的高度为（n-1）时的情况。在这篇报告中，我们研究了平衡重组，当B有高度[log n]。本报告中的两个主要结果是平衡重组的α优化和β优化的NP-困难。第一个结果是由一个序列的多项式时间减少从最小二分法的图（已知是NP-硬），和第二个由一个序列的多项式时间减少从团覆盖的图（已知是NP-硬）。
英文摘要：
The reassembling of a simple connected graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. The reassembling process has a simple formulation (there are several equivalent formulations) relative to a binary tree B (reassembling tree), with root node at the top and $n$ leaf nodes at the bottom, where every cross-section corresponds to a partition of V such that: - the bottom (or first) cross-section (all the leaves) is the finest partition of V with n one-vertex blocks, - the top (or last) cross-section (the root) is the coarsest partition with a single block, the entire set V, - a node (or block) in an intermediate cross-section (or partition) is the result of merging its two children nodes (or blocks) in the cross-section (or partition) below it. The maximum edge-boundary degree encountered during the reassembling process is what we call the alpha-measure of the reassembling, and the sum of all edge-boundary degrees is its beta-measure. The alpha-optimization (resp. beta-optimization) of the reassembling of G is to determine a reassembling tree B that minimizes its alpha-measure (resp. beta-measure). There are different forms of reassembling. In an earlier report, we studied linear reassembling, which is the case when the height of B is (n-1). In this report, we study balanced reassembling, when B has height [log n]. The two main results in this report are the NP-hardness of alpha-optimization and beta-optimization of balanced reassembling. The first result is obtained by a sequence of polynomial-time reductions from minimum bisection of graphs (known to be NP-hard), and the second by a sequence of polynomial-time reductions from clique cover of graphs (known to be NP-hard).
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