An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems

We investigate the computational complexity of the Densestk-Subgraph problem, where the input is an undirected graph G=(V,E) and one wants to find a subgraph on exactly k vertices with the maximum number of edges. We extend previous work on Densestk-Subgraph by studying its parameterized complexity for parameters describing the sparseness of the input graph and for parameters related to the solution size k. On the positive side, we show that, when fixing some constant minimum density μ of the sought subgraph, Densestk-Subgraph becomes fixed-parameter tractable with respect to either of the parameters maximum degree of G and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for Densestk-Subgraph with respect to the combined parameter "degeneracy of G and |V|-k". On the negative side, we find that Densestk-Subgraph is W[1]-hard with respect to the combined parameter "solution size k and degeneracy of G". We furthermore strengthen a previous hardness result for Densestk-Subgraph (Cai, 2008) by showing that for every fixed μ, 0<μ<1, the problem of deciding whether G contains a subgraph of density at least μ is W[1]-hard with respect to the parameter |V|-k. Our positive results are obtained by an algorithmic framework that can be applied to a wide range of Fixed-Cardinality Optimization problems.