Graph Clustering in All Parameter Regimes
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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 λ∈ (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 λ 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 λ? 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 λ∈ (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+ε)-approximate solution in all parameter regimes; we demonstrate simple graph classes for which these bounds are tight.
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