图联盟结构生成
Graph Coalition Structure Generation
摘要：
首先分析了图上的\it联盟结构生成算法的计算复杂性。给定无向图$G=(N,E)$和节点子集上的赋值函数$v:2^N\rightarrow\RR$，问题是找到$N$到连通子集的划分，该划分最大化组件值的总和。这个问题一般是NP-完全的，特别是，它是很难为一个定义的类的估值函数是{ \it独立的断开成员}-也就是说，两个节点没有影响对方的边际贡献，他们的顶点分隔符。尽管如此，对于所有这样的功能，我们提供了一般的和未成年人的自由图的联盟结构生成的复杂性的界限。我们的证明是建设性的，并产生算法解决相应的问题。此外，我们推导出多项式时间界的无圈，$K_{2,3}$和$K_4$小自由图。然而，正如我们所展示的，这个问题仍然是NP-完全的平面图，因此，对于任何$K_k$未成年人的自由图，其中$k\geq 5$。此外，我们的硬度结果保持一个特定的子类的赋值函数，称为{ \it边和}，其中每个子集的节点的值是简单地由给定的权重的总和在诱导子图的边缘。
英文摘要：
We give the first analysis of the computational complexity of {\it coalition structure generation over graphs}. Given an undirected graph $G=(N,E)$ and a valuation function $v:2^N\rightarrow\RR$ over the subsets of nodes, the problem is to find a partition of $N$ into connected subsets, that maximises the sum of the components' values. This problem is generally NP--complete; in particular, it is hard for a defined class of valuation functions which are {\it independent of disconnected members}---that is, two nodes have no effect on each other's marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive polynomial time bounds for acyclic, $K_{2,3}$ and $K_4$ minor free graphs. However, as we show, the problem remains NP--complete for planar graphs, and hence, for any $K_k$ minor free graphs where $k\geq 5$. Moreover, our hardness result holds for a particular subclass of valuation functions, termed {\it edge sum}, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph.
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