Faster Cluster Decomposition with Stronger Bi-Criteria Guarantee

Abstract

Partitioning the vertex set of a graph $G=(V,E)$ into clusters is a classical problem in computer science. As vertices within clusters should be well connected and cuts between clusters sparse, Oveis Gharan and Trevisan (SODA 2014, [OGT14]) proposed to measure the partition quality using inner and outer conductance, i.e., the bi-criteria guarantee. Specifically, when partitioning $V$ into vertex-disjoint subsets $P_1,\ldots, P_k$, every subgraph induced by $P_i$ should have high conductance, while the conductance of $P_i$ in $G$ should be low. They studied the existence of such a partition and designed a polynomial-time algorithm achieving this guarantee. For more than a decade, their algorithm remained the only one achieving this guarantee on its output clusters. In this paper, we study faster cluster decomposition algorithms under the same setup as [OGT14] and prove three results: 1. We make their existential result algorithmic: Our algorithms run in nearly-linear time, and achieve the same bi-criteria guarantee as the one in [OGT14] up to constant factors, including the asymptotically tight dependence on $k$. 2. We prove that their existential result holds algorithmically under a much weaker condition on the initial disjoint subsets. To achieve this, we develop a general algorithm framework that boosts the approximation guarantee of a solution: given an $α$-approximate solution to the $k$-way conductance problem, if $ρ_G(k+1)/ρ_G(k)=Ω\left(α/\varepsilon\right)$, then a $(1+\varepsilon)$-approximate solution can be obtained using only $O(k(1+\logα))$ minimum $s$-$t$ cut computations. 3. We present a nearly-linear time algorithm that finds clusters with the bi-criteria guarantee. Our algorithm has the same guarantee as the one in [OGT14], which, for any unweighted graph of $n$ vertices and $m$ edges, runs in $\widetilde{O}(knm^2)$ time.