Efficient Reassembling of Graphs, Part 2: The Balanced Case

The reassembling of a simple connected graph G = (V,E) with n = ∣V ∣ ⩾ 1 vertices 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 – its so-called 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 (a block in the partition is a node in the cross-section) such that: • the bottom (or first) cross-section (i.e., all the leaves) is the finest partition of V with n one-vertex blocks, • the top (or last) cross-section (i.e., 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 edge-boundary degree of a block A of vertices is the number of edges with one endpoint in A and one endpoint in (V − A). The maximum edge-boundary degree encountered during the reassembling process is what we call the α-measure of the reassembling, and the sum of all edge-boundary degrees is its β-measure. The α-optimization (resp. β-optimization) of the reassembling of G is to determine a reassembling tree B that minimizes its α-measure (resp. β-measure). There are different forms of reassembling, depending on the shape of the reassembling tree B. 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 ⌈logn⌉. In a forthcoming report, we study general reassembling, which is the case when the height of B can be any number between (n − 1) and ⌈logn⌉. The two main results in this report are the NP-hardness of α-optimization and β-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).