Graph Partitioning with Demands: Generalized Conductance and its Applications

2026-07-14

AI Summary

This work introduces and systematically studies the generalized conductance minimization problem: given a graph with vertex demands and edge capacities, find a cut that minimizes generalized conductance while ensuring the total demand in each connected component does not exceed a prescribed upper bound. The authors design a combinatorial approximation algorithm via a two-stage reduction—first to a generalized $k$-multicut problem, then to a sparsest cut problem with demand constraints. Their main contributions include establishing novel connections between generalized conductance and classical graph partitioning problems, extending the framework to demand-aware graph partitioning and hierarchical clustering, obtaining an $O(\log n)$-approximation algorithm, improving this to $O(\sqrt{\log n})$ for multiplicative demand functions, and achieving a constant-factor approximation on trees.

Abstract

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.

Problem

• Graph Partitioning
• Demands
• Generalized Conductance
• Sparsest Cut
• Hierarchical Clustering

Innovation

• generalized conductance
• graph partitioning with demands
• approximation algorithm
• sparsest cut
• hierarchical clustering