A 2-approximation for the bounded treewidth sparsest cut problem in \(\textsf{FPT}\) Time

In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta et al. (ACM STOC, 2013) obtained a 2-approximation with polynomial running time for fixed k, and it remained open the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in \(\textsf{FPT}\) $\mathsf{FPT}$ time. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in \(\textsf{FPT}\) $\mathsf{FPT}$ time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in \(\textsf{FPT}\) $\mathsf{FPT}$ time.