Clustering to Given Connectivities

TL;DR

This paper introduces a new graph clustering problem based on connectivity, proves it is fixed parameter tractable, and provides an algorithm using recursive understanding that handles arbitrary connectivity demands.

Contribution

It defines a novel clustering problem with connectivity constraints and shows it can be efficiently solved when parameterized by the number of edge removals.

Findings

The problem is fixed parameter tractable with respect to k.

An algorithm based on recursive understanding is developed.

The approach handles arbitrary connectivity requirements in the clustering.

Abstract

We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In {\sc Clustering to Given Connectivities}, we are given an $n$-vertex graph $G$, an integer $k$, and a sequence $\Lambda=\langle \lambda_{1},\ldots,\lambda_{t}\rangle$ of positive integers and we ask whether it is possible to remove at most $k$ edges from $G$ such that the resulting connected components are {\sl exactly} $t$ and their corresponding edge connectivities are lower-bounded by the numbers in $\Lambda$. We prove that this problem, parameterized by $k$, is fixed parameter tractable i.e., can be solved by an $f(k)\cdot n^{O(1)}$-step algorithm, for some function $f$ that depends only on the parameter $k$. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that, in out setting, we do not impose any restriction to the connectivity demands in $\Lambda$.