稀疏图和稠密图上的稠密图划分
Dense Graph Partitioning on sparse and dense graphs
摘要：
本文研究了将一个图划分为非固定数目的最大密度的非重叠子图的问题。划分的密度是子图的密度之和，其中子图的密度是它的平均度，即它的边数与它的顶点数之比。这个问题被称为稠密图划分，在一般图上是NP困难的，在树上是多项式时间可解的，并且是多项式时间2-近似的。在本文中，我们研究了稠密图划分的限制，特定的稀疏和稠密图类。特别是，我们证明了它是NP-困难的稠密二部图以及立方图。在$n$顶点上的稠密图上，它在最小度$n-3$上是多项式时间可解的，在$(n-4)$ -正则图上是NP-困难的。我们证明了它在三次图上是多项式时间$4/3$ -可逼近的，并且对于任意常数$t\geq 4$，在最小度$n-t$的图上存在一个有效的多项式时间逼近方案。
英文摘要：
We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on $n$ vertices, it is polynomial-time solvable on graphs with minimum degree $n-3$ and NP-hard on $(n-4)$-regular graphs. We prove that it is polynomial-time $4/3$-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree $n-t$ for any constant $t\geq 4$.
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