Multi-way sparsest cut problem on trees with a control on the number of parts and outliers

Given a graph, the sparsest cut problem asks for a subset of vertices whose edge expansion (the normalized cut given by the subset) is minimized. In this paper, we study a generalization of this problem seeking for $ k $ disjoint subsets of vertices (clusters) whose all edge expansions are small and furthermore, the number of vertices remained in the exterior of the subsets (outliers) is also small. We prove that although this problem is $ NP-$hard for trees, it can be solved in polynomial time for all weighted trees, provided that we restrict the search space to subsets which induce connected subgraphs. The proposed algorithm is based on dynamic programming and runs in the worst case in $ O(k^2 n^3) $, when $ n $ is the number of vertices and $ k $ is the number of clusters. It also runs in linear time when the number of clusters and the number of outliers is bounded by a constant.