Finding Highly Connected Subgraphs
Lecture Notes in Computer Science
A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We show that the problem is NP-hard. Thus, we explore possible parameterizations, such as the solution size, number of vertices in the input, the size of a vertex cover in the input, and the number of edges
... g from the solution (edge isolation). For some parameters, we find strong intractability results; among the parameters yielding tractability, the edge isolation seems to provide the best trade-off between running time bounds and a small value of the parameter in relevant instances. it has to be superpolynomial unless P = NP), and that the parameter value p can be expected to be relatively small in interesting instances. Clearly, there is a trade-off between these goals. Results. We list the results going from the hardest parameters to the easiest, corresponding roughly to going from small expected parameter values to large ones. Let n be the number of vertices in G. For the parameter := n − k (the number of vertices to delete to obtain a highly connected subgraph), we obtain a strong hardness result: there is a trivial n O( ) time algorithm, but it is unlikely that n o( ) time can be achieved (Theorem 1). For the size of the solution k, a fixedparameter algorithm is unlikely, even if we additionally consider the degeneracy of G as a parameter (Theorem 2). If we take the minimum size of a vertex cover τ for G as parameter, we obtain the first fixed-parameter algorithm: the problem can be solved in O((2τ ) τ · n O(1) ) time (Corollary 1). Considering the number of vertices n, we can clearly solve the problem in 2 n ·n O(1) time; however, it is unlikely that this can be improved to 2 o(n) · n O(1) , that is, there is no subexponential-time algorithm (Corollary 2). If the parameter is the number γ of edges between G[S] and the remaining vertices, then we can also find a fixed-parameter algorithm with running time O(4 γ n 2 ) (Theorem 4). Finally, if we consider the number α of edges to delete to obtain a highly connected subgraph (plus singleton vertices), we even obtain a subexponential running time (Theorem 7). Related work. The algorithm by Hartuv and Shamir  partitions a graph heuristically into highly connected components; another algorithm tries to explicitly minimize the number of edges that need to be deleted for this  . Highly connected graphs can be seen as clique relaxation  , that is, a graph class that has many properties similar to cliques, without being as restrictive. Highly connected graphs are very similar to 0.5-quasi-complete graphs  , that is, graphs where every vertex has degree at least (n − 1)/2. Recently, also the task of finding subgraphs with high vertex connectivity has been examined . Preliminaries. We consider only undirected graphs G = (V, E) with n := |V | and m := |E|. We use G − S as shorthand for G[V \ S]. A cut (A, B) in a graph G = (V, E) is a vertex bipartition, that is, A ∩ B = ∅ and A ∪ B = V . The cut edges are the edges with one endpoint in A and one in B, and the size of a cut is the number of its cut edges. For the definitions of FPT, W, parameterized reduction, and problem kernelization refer to  .