Finding Dense Subgraphs of Sparse Graphs

Abstract

Abstract. We investigate the computational complexity of the Densestk-Subgraph (DkS) problem, where the input is an undirected graph G = (V, E) and one wants to find a subgraph on exactly k vertices with a maximum number of edges. We extend previous work on DkS by studying its parameterized complexity. On the positive side, we show that, when fixing some constant minimum density µ of the sought subgraph, DkS becomes fixed-parameter tractable with respect to either of the parameters maximum degree and h-index of G. Furthermore, we obtain a fixedparameter algorithm for DkS with respect to the combined parameter "degeneracy of G and |V | − k". On the negative side, we find that DkS 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 DkS [Cai, Comput. J., 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.

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

“…✩ A preliminary version of this work is contained in the Proceedings of the 7th International Symposium on Parameterized and Exact Computation (IPEC '12) under the title Finding Dense Subgraphs of Sparse Graphs (Komusiewicz and Sorge, 2012) [36]. The current article includes improved running times as well as generalized algorithmic results.…”

“…A rather simple direct analysis of Γ proves that it contains at most O((4(∆ − 1)) k ) leaves [36]. In order to prove the better bound stated in Theorem 2, we divide the nodes of Γ into different types.…”

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Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists

“…We are given a graph G and a positive integer k. The task is to find an induced subgraph of G with k vertices that contains the maximum number of edges. It is known that DkS is W[1]-hard with parameter k [5]. We give an FPT-reduction from DkS to MMI(k).…”

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