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تسجيل مستخدم جديدEfficient Reassembling of Graphs, Part 1: The Linear Case
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The reassembling of a simple connected graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps: (1) We cut every edge of G into two halves, thus obtaining a collection of n=|V| one-vertex components. (2) We splice the two halves of every edge together, not of all the edges at once, but in some ordering Theta of the edges that minimizes two measures that depend on the edge-boundary degrees of assembled components. The edge-boundary degree of a component A (subset of V) is the number of edges in G with one endpoint in A and one endpoint in V-A. We call the maximum edge-boundary degree encountered during the reassembling process 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 an order Theta for splicing the edges that minimizes its alpha-measure (resp. beta-measure). There are different forms of reassembling. We consider only cases satisfying the condition that if the an edge between disjoint components A and B is spliced, then all the edges between A and B are spliced at the same time. In this report, we examine the particular case of linear reassembling, which requires that the next edge to be spliced must be adjacent to an already spliced edge. We delay other forms of reassembling to follow-up reports. We prove that alpha-optimization of linear reassembling and minimum-cutwidth linear arrangment (CutWidth) are polynomially reducible to each other, and that beta-optimization of linear reassembling and minimum-cost linear arrangement (MinArr) are polynomially reducible to each other.
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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 t
ree 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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