The price of connectedness in graph partitioning problems

Abstract

In this work, we present some results coming from the comparison of two classes of problems of optimal graph partitioning. In the first class of problem, called further constrained problem, ones impose the nodes belonging to a same cluster to be connected. In the second class of problem, no constraint of connectedness is imposed. Obviously, the optimal solution of an unconstrained problem is always better than or equal to the solution of the corresponding constrained problem. It is thus interesting to estimate how much the connectedness constraint degrades the optimal solution of the partitioning problem. This degradation is what we call here the price of connectedness. Motivated by a numerical example, we propose to estimate the price of connectedness by comparing the cardinalities of the feasible sets in the unconstrained and constrained optimisation problems. We present then a tight upper-bound on the ratio of these two cardinalities for a directed Erdős-Renyi graph. On our way to set this result we also derive an upper-bound on the probability that an Erdős-Renyi graph is connected and the exact probability that an Erdős-Renyi graph contains isolated nodes.