权值不相关的非均匀图划分
Nonuniform Graph Partitioning with Unrelated Weights
摘要：
我们给出了最小非均匀划分问题的双准则近似算法，最近由Krauthgamer，Naor，Schwartz和Talwar（2014）介绍。在这个问题中，我们给出了一个图$G=(V,E)$上的$n$顶点和$k$数$ρ_1,\dots, ρ_k$。目标是将图划分为满足$|P_i|\leq ρ_i n$的$k$不交集$P_1,\dots, P_k$，以使划分所切割的边数最小。我们的算法具有$O(\sqrt{\log n \log k})$的一般图的近似比，和一个$O(1)$近似的排除未成年人的图。这是对Krauthgamer、Naor、Schwartz和Talwar（2014）的$O(\log n)$算法的改进。我们的近似比匹配最小（均匀）$k$ -分区问题的最佳已知比。我们扩展我们的结果的情况下，“无关的权重”和“无关的$d$维权重”的情况下。在前一种情况下，不同的顶点可以具有不同的权重，并且顶点的权重可以取决于顶点被分配到的集合$P_i$。在后一种情况下，如果$u$被分配给$P_i$，则每个顶点$u$具有$d$维权重$r(u,i) = (r_1(u,i), \dots, r_d(u,i))$。每个集合$P_i$具有$d$维容量$c(i) = (c_1(i),\dots, c_d(i))$。目标是找到一个分区，使得$\sum_{u\in {P_i}} r(u,i) \leq c(i)$坐标明智的。
英文摘要：
We give a bi-criteria approximation algorithm for the Minimum Nonuniform Partitioning problem, recently introduced by Krauthgamer, Naor, Schwartz and Talwar (2014). In this problem, we are given a graph $G=(V,E)$ on $n$ vertices and $k$ numbers $ρ_1,\dots, ρ_k$. The goal is to partition the graph into $k$ disjoint sets $P_1,\dots, P_k$ satisfying $|P_i|\leq ρ_i n$ so as to minimize the number of edges cut by the partition. Our algorithm has an approximation ratio of $O(\sqrt{\log n \log k})$ for general graphs, and an $O(1)$ approximation for graphs with excluded minors. This is an improvement upon the $O(\log n)$ algorithm of Krauthgamer, Naor, Schwartz and Talwar (2014). Our approximation ratio matches the best known ratio for the Minimum (Uniform) $k$-Partitioning problem. We extend our results to the case of "unrelated weights" and to the case of "unrelated $d$-dimensional weights". In the former case, different vertices may have different weights and the weight of a vertex may depend on the set $P_i$ the vertex is assigned to. In the latter case, each vertex $u$ has a $d$-dimensional weight $r(u,i) = (r_1(u,i), \dots, r_d(u,i))$ if $u$ is assigned to $P_i$. Each set $P_i$ has a $d$-dimensional capacity $c(i) = (c_1(i),\dots, c_d(i))$. The goal is to find a partition such that $\sum_{u\in {P_i}} r(u,i) \leq c(i)$ coordinate-wise.
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