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

Tobias Mömke
23rd Conference on Integer Programming and Combinatorial Optimization (IPCO'22) (2022)
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In the non-uniform sparsest cut problem, we are given a supply graph $G$ and a demand graph $D$, both with the same vertex set $V$. The goal is to partition the set of vertices $V$ into two subsets in order to minimize the ratio of the total capacity on the supply edges crossing from one part to the other over the total demand of the crossing edges. In this work, we study the 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 it remained open the question of whether there exists a $c$-approximation algorithm for a constant $c$ independent of $k$ that runs in FPT time. We answer the above question in the affirmative. We show that there exists a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in FPT time. 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 construction, we construct a relaxation driven by a tree decomposition of the supply graph, and we include a carefully chosen set of lifting variables to encode information of subsets of nodes with super-constant size, while at the same time having a sufficiently small linear program that can be solved in FPT time.