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

TL;DR

This paper investigates the parameterized complexity of the $k$-Sparsest Cut and Small-Set Expansion problems, providing complexity classifications and fixed-parameter tractability results based on various graph parameters.

Contribution

It establishes NP-hardness and W[1]-hardness results, and identifies conditions under which these problems are fixed-parameter tractable.

Findings

Weighted $k$-Sparsest Cut is NP-hard for all $k \\geq 3$ on graphs with bounded vertex cover.

Unweighted $k$-Sparsest Cut is W[1]-hard when parameterized by combined graph parameters.

Unweighted $k$-Sparsest Cut is FPT when parameterized by vertex cover number and $k$, and by treewidth when $k$ is fixed.

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.