The Complexity Status of Problems Related to Sparsest Cuts

Given an undirected graph G = (V, E) with a capacity function w : E -> Z(+) on the edges, the sparsest; cut problem is to find a vertex subset S subset of V minimizing Sigma(e epsilon E(S,V\S)) w(e)/(vertical bar S vertical bar vertical bar\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 is an element of E) the problem is to minimize vertical bar E(S, V \ S)vertical bar/(vertical bar S vertical bar vertical bar V \ S vertical bar) over all subsets S subset of 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.