Mean isoperimetry with control on outliers: Exact and approximation algorithms

Abstract

Given a weighted graph G=(V,E) with weight functions c:E→R+ and π:V→R+, and a subset U⊆V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of vertices in U. The mean isoperimetry problem, ISO1(G,k), for a weighted graph G is a generalization of the classical uniform sparsest cut problem in which, given a parameter k, the objective is to find k disjoint nonempty subsets of V minimizing the average normalized cut value of the parts. The robust version of the problem seeks an optimizer where the number of vertices that fall out of the subpartition is bounded by some given integer 0≤ρ≤|V|. The problem may also be considered as the normalized version of the classical k-multiway cut problem on edge-weighted graphs.Our main result states that ISO1(G,k), as well as its robust version, CRISO1(G,k,ρ), subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time O(k2ρ2π(V(T))3) on any weighted tree T, in which π(V(T)) is the sum of the vertex-weights. This result implies that ISO1(G,k) is strongly polynomial-time solvable on weighted trees when the vertex-weights are polynomially bounded and may be compared to the fact that the problem is NP-hard for weighted trees in general. As far as applications are concerned, the connectivity requirement may be interpreted as an approach to model the practical consistency of the parts, which together with having control on the size of the outlier set and applying a smooth “mean” cost function (as opposed to, say, the “max” version), characterizes our solution to CRISO1(G,k,ρ) on trees as one of the most flexible and accurate procedures within the framework of isoperimetry-based clustering.Also, using this, we show that both mentioned problems, ISO1(G,k) and CRISO1(G,k,ρ) as well as the ordinary robust mean isoperimetry problem RISO1(G,k,ρ), admit polynomial-time O(log1.5⁡|V|log⁡log⁡|V|)-approximation algorithms for weighted graphs with polynomially bounded weights, using the Räcke-Shah tree cut sparsifier.