Parameterized Algorithms for Graph Partitioning Problems

We study a broad class of graph partitioning problems. Each problem is defined by two constants, α ₁ and α ₂. The input is a graph G, an integer k and a number p, and the objective is to find a subset \(U\subseteq V\) of size k, such that α ₁ m ₁ + α ₂ m ₂ is at most (or at least) p, where m ₁, m ₂ are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k-Sparsest Subgraph. Our main result is a 4 ^{k + o(k)}Δ ^k ⋅n ^O(1) time algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. (Proc. International Symposium on Parameterized and Exact Computation, 2013). We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give a 4 ^{p + o(p)}⋅n ^O(1) time algorithm for Max (k, n − k)-Cut, thus improving significantly the best known p ^p ⋅n ^O(1) time algorithm by Bonnet et al.