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

We study the NP-hard 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 S ⊆ V 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 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 k ≥ 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. On the negative side, first we prove that for every fixed integer k,τ≥ 3, the problem kSC is NP-hard for graphs with vertex cover number at most τ. 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 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.

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