The Complexity Status of Problems Related to Sparsest Cuts

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Given an undirected graph G = (V,E) with a capacity function \(w : E \longrightarrow \mathbb{Z}^+\) on the edges, the sparsest cut problem is to find a vertex subset S ⊂ V minimizing ∑  e ∈ E(S,V ∖ S) w(e)/(|S||V ∖ S|). This problem is NP-hard. The proof can be found in [16]. In the case of unit capacities (i. e. if w(e) = 1 for every e ∈ E) the problem is to minimize |E(S,V ∖ S)|/(|S||V ∖ S|) over all subsets S ⊂ V. While this variant of the sparsest cut problem is often assumed to be NP-hard, this note contains the first proof of this fact. We also prove that the problem is polynomially solvable for graphs of bounded treewidth.