带需求的图划分：广义电导及其应用

中文摘要 AI 辅助

在这项工作中，我们研究了一般需求模型下的各种图划分问题。在每个此类任务中，给定一个图\(G=(V,E,c,w)\)，其中有容量函数\(c\colon E\to \mathbb{N}\)和需求函数\(w\colon V\times V\to \mathbb{N}\)。我们主要关注找到一个割\((S, \bar{S})\)，使量\(\psi_w( S ) = \frac{c( S, \bar{S} )}{w( S, V )\cdot w( \bar{S}, V )}\)最小的问题。这里，\(c( S, \bar{S} )\)是\(S\)与其补集\(\bar{S}\)之间边的成本，\(w( S, V )=w( S )+w( S, \bar{S} )\)是\(S\)内的内部需求\(w( S )\)与\(S\)和\(\bar{S}\)顶点之间需求\(w( S, \bar{S} )\)的总和。我们称\(\psi_w( S )\)为割\((S, \bar{S})\)的广义电导，最小化\(\psi_w( S )\)的任务为广义电导问题。我们的主要贡献是为此目标提供一个具有\(\mathcal{O}(\log n)\)近似保证的算法。我们的结果通过双向归约实现：首先归约到著名的广义\(k -\)多割问题，然后归约到经典稀疏割问题的一个受限变体，对可割的需求量有额外的上界约束。此外，我们表明上述过程可用于获得带需求的图划分的\(\mathcal{O}(\log n)\)双标准近似，目标是找到边的最小成本子集\(C\)，使得对于\(G\setminus C\)的每个组件\(H\)，\(w( H )\leq \rho\cdot w( V )\)。这进而为带需求的层次聚类产生一个\(\mathcal{O}(\log n)\)近似，即找到将图划分为越来越精细簇的割层次结构的问题。对于乘法需求函数，我们将这些保证改进到\(\mathcal{O}(\sqrt{\log n})\)，对于树，我们对所有目标都得到一个\(\mathcal{O}(1)\)近似。

英文摘要

In this work, we study various graph partitioning problems under a general demand model. In each such task, we are given a graph $G=(V,E,c,w)$ with a capacity function $c\colon E\to \mathbb{N}$ and a demand function $w\colon V\times V\to \mathbb{N}$. Our main focus is the problem of finding a cut $(S, \bar{S})$ minimizing the quantity \[ ψ_w( S ) = \frac{c( S, \bar{S} )}{w( S, V )\cdot w( \bar{S}, V )}. \] Here, $c( S, \bar{S} )$ is the cost of edges between $S$ and the complement of $S$, $\bar{S}$, and $w( S, V )=w( S )+w( S, \bar{S} )$ is the sum of the internal demand within $S$, $w( S )$, and the demand between vertices of $S$ and $\bar{S}$, $w( S, \bar{S} )$. We call $ψ_w( S )$ the \emph{generalized conductance} of the cut $(S, \bar{S})$, and the task of minimizing $ψ_w( S )$ the Generalized Conductance Problem. Our main contribution is an algorithm with an $\mathcal{O}(\log n)$-approximation guarantee for this objective. Our result is achieved via a two-way reduction: first to the well-known Generalized $k$-Multicut Problem, and then to a constrained variant of the classic Sparsest-Cut Problem, with an additional upper-bound constraint on the amount of demand that may be cut. Moreover, we show that the above procedure can be used to obtain an $\mathcal{O}(\log n)$-bicriteria approximation for Graph Partitioning with Demands, where the goal is to find a minimum-cost subset of edges $C$ such that for every component $H$ of $G\setminus C$, $w( H )\leq ρ\cdot w( V )$. This, in turn, yields an $\mathcal{O}(\log n)$-approximation for Hierarchical Clustering with Demands, the problem of finding a hierarchy of cuts that partitions the graph into increasingly refined clusters. For multiplicative demand functions, we improve these guarantees to $\mathcal{O}(\sqrt{\log n})$ and for trees we get an $\mathcal{O}(1)$-approximation for all of our objectives.

补充信息

备注

24 pages