Graph Clustering in All Parameter Regimes

TL;DR

This paper develops a polynomial-time method to find a small family of clusterings that approximately optimize a parameterized graph clustering objective across all resolution parameters, unifying multiple clustering regimes.

Contribution

It introduces a logarithmic family of clusterings that approximate the LambdaPrime objective for all parameters, with proven tight bounds and practical algorithms.

Findings

Logarithmic number of clusterings suffices for approximation across all parameters.

Tight bounds established for ring graphs and general graphs.

Existence of small clustering families for exact and approximate solutions.

Abstract

Resolution parameters in graph clustering represent a size and quality trade-off. We address the task of efficiently solving a parameterized graph clustering objective for all values of a resolution parameter. Specifically, we consider an objective we call LambdaPrime, involving a parameter $\lambda \in (0,1)$. This objective is related to other parameterized clustering problems, such as parametric generalizations of modularity, and captures a number of specific clustering problems as special cases, including sparsest cut and cluster deletion. While previous work provides approximation results for a single resolution parameter, we seek a set of approximately optimal clusterings for all values of $\lambda$ in polynomial time. In particular, we ask the question, how small a family of clusterings suffices to optimize -- or to approximately optimize -- the LambdaPrime objective over the full possible spectrum of $\lambda$? We obtain a family of logarithmically many clusterings by solving the parametric linear programming relaxation of LambdaPrime at a logarithmic number of parameter values, and round their solutions using existing approximation algorithms. We prove that this number is tight up to a constant factor. Specifically, for a certain class of ring graphs, a logarithmic number of feasible solutions is required to provide a constant-factor approximation for the LambdaPrime LP relaxation in all parameter regimes. We additionally show that for any graph with $n$ nodes and $m$ edges, there exists a set of $m$ or fewer clusterings such that for every $\lambda \in (0,1)$, the family contains an exact solution to the LambdaPrime objective. There also exists a set of $O(\log n)$ clusterings that provide a $(1+\varepsilon)$-approximate solution in all parameter regimes; we demonstrate simple graph classes for which these bounds are tight.