A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

Abstract

There has been intensive work on the parameterized complexity of the typically NPhard task to edit undirected graphs into graphs fulfilling certain given vertex degree constraints. In this work, we lift the investigations to the case of directed graphs; herein, we focus on arc insertions. To this end, we develop a general two-stage framework which consists of efficiently solving a problem-specific number problem and transferring its solution to a solution for the graph problem by applying flow computations. In this way, we obtain fixed-parameter tractability and polynomial kernelizability results, with the central parameter being the maximum vertex in-or outdegree of the output digraph. Although there are certain similarities with the much better studied undirected case, the flow computation used in the directed case seems not to work for the undirected case while f -factor computations as used in the undirected case seem not to work for the directed case.

A survey of parameterized algorithms and the complexity of edge modification

“…Bredereck et al [40] extended some results of Froese et al [127], Golovach and Mertzios [136] and Hartung et al [162] for directed graphs. We already mentioned Digraph Degree Constraint Completion in the previous section but, they also considered more general Digraph Degree Constraint Sequence Completion that combines individual degree and degree sequence constraints.…”

“…All aforementioned problems are stated for undirected graphs. The systematic study of the degree constraint modification problems for directed graphs was recently initiated by Bredereck, Froese, Koseler, Millani, Nichterlein and Niedermeier [40]. We will return to this paper in the next section where we consider degree sequence restriction, but here we mention only that they considered the Digraph Degree Constraint Completion problem that could be seen as a variant of DCE(S), where for each vertex, a degree list function that assigns to each vertex a set of pairs of non-negative integers from {0, .…”

“…, r} that specify the desired pairs of values of in-and out-degrees respectively are given and S = {edge (arc) addition}. Bredereck et al [40] show that this problem admits a kernel with O(r 5 ) vertices.…”

Towards publishing directed social network data with k‐degree anonymization

“…Backstrom et al 22 show that removing the attribute of username is not sufficient to protect users from the attack; with certain background knowledge, the adversary is still able to obtain sensitive information. In addition, such background knowledge can be easily obtained from personal social network accounts or some other related websites.…”

“…This example emphasizes the need to transform the original social network dataset $$$ G $$$ to a dataset $$$ {G}^{\prime } $$$ before publication so as to avoid the association of specific records to particular persons. Backstrom et al 22 show that removing the attribute of username is not sufficient to protect users from the attack; with certain background knowledge, the adversary is still able to obtain sensitive information. In addition, such background knowledge can be easily obtained from personal social network accounts or some other related websites.…”