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 \(\mathsf {FPT}\) 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 \(\mathsf {FPT}\) 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 \(\mathsf {FPT}\) time.
Partially supported by DFG Grant 439522729 (Heisenberg-Grant), DFG Grant 439637648 (Sachbeihilfe), and ANID Grants ACT210005 and FONDECYT 11190789.
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Notes
- 1.
A full version of this article is available in Arxiv and can be found in [13].
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Cohen-Addad, V., Mömke, T., Verdugo, V. (2022). A 2-Approximation for the Bounded Treewidth Sparsest Cut Problem in \(\mathsf {FPT}\) Time. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_9
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