Degree-Constrained Editing of Small-Degree Graphs

This thesis deals with degree-constrained graph modification problems. In particular, we investigate the computational complexity of DAG Realization and Degree Anonymity. The DAG Realization problem is, given a multiset of positive integer pairs, to decide whether there is a realizing directed acyclic graph (DAG), that is, pairs are one-to-one assigned to vertices such that the indegree and the outdegree of every vertex coincides with the two integers of the assigned pair. The Degree Anonymity problem is, given an undirected graph G and two positive integers k and s, to decide whether at most s graph modification operations can be performed in G in order to obtain a k-anonymous graph, that is, a graph where for each vertex there are k ? 1 other vertices with the same degree.[–]
We classify both problems as NP-complete, that is, there are presumably no polynomial-time algorithms that can solve every instance of these problems. Confronted with this worst-case intractability, we perform a parameterized complexity study in order to detect efficiently solvable special cases that are still practically relevant. The goal herein is to develop fixed-parameter algorithms where the seemingly unavoidable exponential dependency in the running time is confined to a parameter of the input. If the parameter is small, then the corresponding fixed-parameter algorithm is fast. The parameter thus measures some structure in the input whose exploitation makes the particular input tractable. Considering Degree Anonymity, two natural parameters provided with the input are anonymity level k and solution size s. However, we will show that Degree Anonymity is W[1]-hard with respect to the parameter s even if k = 2. This means that the existence of fixed-parameter algorithms for s and k is very unlikely. Thus, other parameters have to be considered. We will show that the parameter maximum vertex degree is very promising for both DAG Realization and Degree Anonymity. Herein, for Degree Anonymity, we consider the maximum degree of the input graph. Considering DAG Realization, we take the maximum degree in a realizing DAG. Due to the problem definition, we can easily determine the maximum degree by taking the maximum over all integers in the given multiset. We provide fixed-parameter algorithms with respect to the maximum degree for DAG Realization and for Anonym E-Ins. The later is the variant of Degree Anonymity when only edge insertions are allowed as modification operations. If we allow edge deletions or vertex deletions as graph modification operations, then we can show that the corresponding variants of Degree Anonymity—called Anonym V-Del and Anonym E-Del—are NP-complete even if the maximum vertex degree is seven. Moreover, we provide strong intractability results for Anonym E-Del and Anonym V-Del proving that they remain NP-complete in several restricted graph classes. Studying the approximability of natural optimization problems associated with Anonym E-Del or Anonym V-Del, we obtain negative results showing that none of the considered problems can be approximated in polynomial time better than within a factor of n^(1/2) where n denotes the number of vertices in the input. Furthermore, for the optimization variants where the solution size s is given and the task is to maximize the anonymity level k, this inapproximability even holds if we allow a running time that is exponential in s.
Observe that DAG Realization also can be seen as degree-constrained graph modification problem where only arc insertions are allowed: Starting with an arcless graph, the task is to insert arcs to obtain a realizing DAG for the given multiset. The above classification with respect to the parameter maximum degree shows that in graphs with small maximum degree the modification operation edge respectively arc insertion is easier than vertex or edge deletion. There is a plausible explanation for this behavior: When the maximum degree is small, then there is a high freedom in inserting edges or arcs as for a given vertex almost all other vertices can be chosen as new neighbor. Observe that for DAG Realization the additional requirement that the directed graph shall be acyclic restricts this freedom. In Anonym E-Ins, we do not have restrictions on this freedom. In fact, exploiting this freedom in our implementation for Anonym E-Ins, we show that our theoretical ideas can be turned into successful heuristics and lower bounds. Experiments on several large-scale real-world datasets show that our implementation significantly improves on a recent heuristic and provides (provably) optimal solutions on about 21 % (56 of 260) of the real-world data.