Finding Highly Connected Subgraphs

Abstract

Abstract. 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 outgoing 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.

Multivariate Algorithmics for Finding Cohesive Subnetworks

“…This might yield efficient algorithms to identify certain types of alliances. A related field would be to systematically investigate the combination of clique relaxations and isolation concepts [24,27,58,68], where the number of outgoing edges from a set S is restricted.…”

“…HIGHLY CONNECTED SUBGRAPH is NP-hard and W[1]-hard for the parameter (d, k) and W [1]-hard for the parameter [58]. The parameter ∆ has not been considered so far, but the fact that every highly connected graph on k vertices has minimum degree k/2 or more implies that the problem is nontrivial only if k ≤ 2∆.…”

The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

“…Furthermore, replacing the bound on the open neighborhood in the case of small secludedness by a bound on the outgoing edges of a solution would be an interesting modification of the problem. The variation follows the idea of the concept of isolation [28,30,31,34]. As the number of outgoing edges is at least as large as the open neighborhood, this might offer new possibilities for fixed-parameter algorithms.…”

“…The concept of isolation states that the solution should have few edges to the rest of the graph and was originally introduced for finding cliques [31]. Isolation was subsequently explored also for more general definitions of dense subgraphs [28,30,31,34]. Chiefly the constraint that the vertices in the solution shall have maximum/minimum/average outdegree bounded by a parameter was considered [28,31,34], leading to various parameterized tractability and hardness results.…”

“…Chiefly the constraint that the vertices in the solution shall have maximum/minimum/average outdegree bounded by a parameter was considered [28,31,34], leading to various parameterized tractability and hardness results. Also the overall number of edges outgoing the solution has been studied recently [30]. Finding isolated vertices without constraint on their topology was already studied by Downey et al [17].…”