Given a weighted graph $G=(V,E)$ with weight functions $c:Eto mathbb{R}_+$ and $pi:Vto mathbb{R}_+$, and a subset $Usubseteq V$, 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 {it mean isoperimetry problem}, $mathsf{ISO}^1(G,k)$, 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 $0 leq rho leq |V|$. Our main result states that $mathsf{ISO}^1(G,k)$, as well as its robust version, $mathsf{CRISO}^1(G,k,rho)$, subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time $O(k^2 rho^2 pi(V(T)^3)$ on any weighted tree $T$, in which $pi(V(T))$ is the sum of the vertex-weights. This result implies that $mathsf{ISO}^1(G,k)$ 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. Also, using this, we show that both mentioned problems, $mathsf{ISO}^1(G,k)$ and $mathsf{CRISO}^1(G,k,rho)$ as well as the ordinary robust mean isoperimetry problem $mathsf{RISO}^1(G,k,rho)$, admit polynomial-time $O(log^{1.5}|V| loglog |V|)$-approximation algorithms for weighted graphs with polynomially bounded weights, using the R{a}cke-Shah tree cut sparsifier.
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