AI 中文总结
研究图划分问题的连通和密集划分变体，通过证明厚森林的最大密集图划分算法、有界树宽图的连通划分动态规划算法，以及分裂图上的\(\mathsf{NP}\)难结果，明确了该问题多项式可计算性的边界。
AI 中文摘要
我们关注图划分问题的两个变体，即连通划分和密集划分。对于图\(G=(V,E)\)及其顶点划分\(\mathcal{P}=\{P_1,\ldots, P_k\}\)，若每个\(P_i\)在\(G\)中诱导出连通图，则\(\mathcal{P}\)是\(G\)的连通划分。许多经典变体对部分数量和每个部分大小都有额外限制。我们定义划分\(\mathcal{P}\)的密度为\(d(\mathcal{P})=\sum_{i=1}^k |E(P_i)|/|V(P_i)|\)。最大密集图划分问题要求构造最大密度的划分。我们研究了固定集合数量\(k\)和不固定\(k\)的情况，证明了：1. 为弦图子类厚森林的最大密集图划分设计多项式时间算法，推广了块图上已知的多项式时间算法。2. 给出通用动态规划算法，在有界树宽图上构造（若可能）规定大小的\(k\)个集合的连通划分，得到密集图划分两个变体的算法及Győri-Lovász定理的有效构造。3. 证明最大密集图划分到\(k\)部分在分裂图上是\(\mathsf{NP}\)难的，表明厚树是该问题多项式可计算性的边界。
英文摘要
We focus on two variants of graph partitioning problems, connected partition and dense partition. Formally, given a graph $G=(V,E)$ and a partition of its vertices $\mathcal P=\{P_1,\ldots, P_k\}$ we say that $\mathcal P$ is a connected partition of $G$ if each $P_i$ induces a connected graph in $G$. Many classical variants of this problem impose additional restrictions both on the number of parts as well as on the size of each part. Moreover, given a partition $\mathcal P=\{P_1,\ldots, P_k\}$ we define its density by $d(\mathcal P):=\sum_{i=1}^k |E(P_i)|/|V(P_i)|$. The problem Maximum Dense Graph Partition asks to construct a partition of maximum density. We study this problem both with and without fixed number of sets $k$. We prove the following results: 1. A polynomial time algorithm for Maximum Dense Graph Partition of thick forests, a subclass of chordal graphs, generalizing the previously known polynomial time algorithm on block graphs. 2. A generic dynamic programming algorithm to construct (if possible) a connected partition into $k$ sets of prescribed sizes on graphs with bounded treewidth. This yields algorithms for both variants of Dense Graph Partition and an efficient construction for the Győri-Lovász theorem. 3. The $\mathsf{NP}$-hardness of Maximum Dense Graph Partition to $k$ parts restricted to split graphs, indicating that thick trees are the boundary for the polynomial computability of this problem.
Comments35 pages, 6 Figures
