A 2-Approximation for the Bounded Treewidth Sparsest Cut Problem in FPT Time

TL;DR

This paper presents a fixed-parameter tractable 2-approximation algorithm for the non-uniform sparsest cut problem in graphs with bounded treewidth, improving computational efficiency for this class of problems.

Contribution

The authors develop a novel FPT-time 2-approximation algorithm for bounded treewidth supply graphs, using a specialized Sherali-Adams relaxation based on tree decompositions.

Findings

Achieved a 2-approximation ratio in FPT time for bounded treewidth graphs.

Introduced a new LP relaxation leveraging tree decompositions and selective lifting variables.

Extended the understanding of approximation algorithms for sparsest cut problems in special graph classes.

Abstract

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, Talwar and Witmer [STOC 2013] obtained a 2-approximation with polynomial running time for fixed k, and the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in FPT time, remained open. 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 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 FPT time.