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

Christian Komusiewicz, Manuel Sorge

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