On the Parameterized Complexity of Sparsest Cut and Small-set Expansion Problems

Ramin Javadi; Amir Nikabadi

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arXiv · 1910.12353

On the Parameterized Complexity of Sparsest Cut and Small-set Expansion Problems

Ramin Javadi·Amir Nikabadi

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Abstract

We present a parameterized dichotomy for the \textsc{$k$-Sparsest Cut} problem in weighted and unweighted versions. In particular, we show that the weighted \textsc{$k$-Sparsest Cut} problem is NP-hard for every $k\geq 3$ even on graphs with bounded vertex cover number. Also, the unweighted \textsc{$k$-Sparsest Cut} problem is W[1]-hard when parameterized by the three combined parameters tree-depth, feedback vertex set number, and $k$. On the positive side, we show that unweighted \textsc{$k$-Sparsest Cut} problem is FPT when parameterized by the vertex cover number and $k$, and when $k$ is fixed, it is FPT with respect to the treewidth. Moreover, we show that the generalized version \textsc{$k$-Small-Set Expansion} problem is FPT when parameterized by $k$ and the maximum degree of the graph, though it is W[1]-hard for each of these parameters separately.

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Ramin Javadi, Amir Nikabadi. 2019-10-27. On the Parameterized Complexity of Sparsest Cut and Small-set Expansion Problems. https://arxiv.org/abs/1910.12353

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