Breaking symmetries to rescue sum of squares in the case of makespan scheduling

Verdugo, Victor; Verschae, José; Wiese, Andreas

doi:10.1007/s10107-020-01511-3

Cited by 3 publications

(1 citation statement)

References 44 publications

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“…The impact of the treewidth parameter has also been studied in the context of polynomial optimization [4]. Finally, we mention that the Sherali-Adams hierarchy has been useful to design algorithms in other minor-free and bounded treewidth graph problems, including independent set and vertex cover [24,5], and also in several recent results on scheduling and clustering [29,16,25,15,1].…”

Section: Related Workmentioning

confidence: 99%

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

Cohen-Addad¹,

Mömke²,

Verdugo³

2021

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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 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.

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