We study the NP-hard \textsc{k-Sparsest Cut} problem (kSC) in which,
given an undirected graph G=(V,E) and a parameter k, the objective is to
partition vertex set into k subsets whose maximum edge expansion is
minimized. Herein, the edge expansion of a subset SβV is defined as
the sum of the weights of edges exiting S divided by the number of vertices
in S. Another problem that has been investigated is \textsc{k-Small-Set
Expansion} problem (kSSE), which aims to find a subset with minimum edge
expansion with a restriction on the size of the subset. We extend previous
studies on kSC and kSSE by inspecting their parameterized complexity. On
the positive side, we present two FPT algorithms for both kSSE and 2SC
problems where in the first algorithm we consider the parameter treewidth of
the input graph and uses exponential space, and in the second we consider the
parameter vertex cover number of the input graph and uses polynomial space.
Moreover, we consider the unweighted version of the kSC problem where kβ₯2 is fixed and proposed two FPT algorithms with parameters treewidth and
vertex cover number of the input graph. We also propose a randomized FPT
algorithm for kSSE when parameterized by k and the maximum degree of the
input graph combined. Its derandomization is done efficiently.
\noindent On the negative side, first we prove that for every fixed integer
k,Οβ₯3, the problem kSC is NP-hard for graphs with vertex cover
number at most Ο. We also show that kSC is W[1]-hard when parameterized
by the treewidth of the input graph and the number~k of components combined
using a reduction from \textsc{Unary Bin Packing}. Furthermore, we prove that
kSC remains NP-hard for graphs with maximum degree three and also graphs with
degeneracy two. Finally, we prove that the unweighted kSSE is W[1]-hard for
the parameter k