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.

翻译：我们针对加权和非加权版本的\textsc{$k$-稀疏割}问题提出了参数化二分法。特别地，我们证明即使在有界顶点覆盖数的图上，加权\textsc{$k$-稀疏割}问题对于每个$k\geq 3$都是NP难的。同时，非加权\textsc{$k$-稀疏割}问题在以树深度、反馈顶点集数和$k$这三个组合参数为参数时是W[1]-难的。在积极方面，我们证明非加权\textsc{$k$-稀疏割}问题在以顶点覆盖数和$k$为参数时是FPT的，且当$k$固定时，该问题关于树宽是FPT的。此外，我们证明推广版本\textsc{$k$-小集合扩张}问题在以$k$和图的最大度为参数时是FPT的，尽管这两个参数单独考虑时该问题都是W[1]-难的。