André Nichterlein

On tractable cases of Target Set Selection

We study the NP-hard Target Set Selection (TSS) problem occurring in social network analysis. Roughly speaking, given a graph where each vertex is associated with a threshold, in TSS the task is to select a minimum-size “target set” such that all vertices of the graph get activated. Activation is a dynamic process. First, only the vertices in the target set are active. Then, a vertex becomes active if the number of its active neighbors exceeds its threshold, and so on. TSS models the spread of information, infections, and influence in networks. Complementing results on its polynomial-time approximability and extending results for its restriction to trees and bounded treewidth graphs, we classify the influence of the parameters “diameter”, “cluster editing number”, “vertex cover number”, and “feedback edge set number” of the underlying graph on the problem’s computational complexity, revealing both tractable and intractable cases. For instance, even for diameter-two split graphs TSS remains W[2]-hard with respect to the parameter “size of the target set”. TSS can be efficiently solved on graphs with small feedback edge set number and also turns out to be fixed-parameter tractable when parameterized by the vertex cover number. Both results contrast known parameterized intractability results for the parameter “treewidth”. While these tractability results are relevant for sparse networks, we also show efficient fixed-parameter algorithms for the parameter “cluster editing number”, yielding tractability for certain dense networks.

On Structural Parameterizations for the 2-Club Problem

The NP-hard 2-Club problem is, given an undirected graph G = (V,E) and a positive integer l, to decide whether there is a vertex set of size at least l that induces a subgraph of diameter at most two. We make progress towards a systematic classification of the complexity of 2-Club with respect to structural parameterizations of the input graph. Specifically, we show NP-hardness of 2-Club on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter three. Moreover, we present an algorithm that solves 2-Club in |V| f(k) time, where k is the so-called h-index of the input graph. By showing W[1]-hardness for this parameter, we provide evidence that the above algorithm cannot be improved to a fixed-parameter algorithm. This also implies W[1]-hardness with respect to the degeneracy of the input graph. Finally, we show that 2-Club is fixed-parameter tractable with respect to “distance to co-cluster graphs” and “distance to cluster graphs”.

The complexity of finding a large subgraph under anonymity constraints

We define and analyze an anonymization problem in undirected graphs, which is motivated by certain privacy issues in social networks. The goal is to remove a small number of vertices from the graph such that in the resulting subgraph every occurring vertex degree occurs many times.

Win-win kernelization for degree sequence completion problems

We study provably effective and efficient data reduction for a class of NP-hard graph modification problems based on vertex degree properties. We show fixed-parameter tractability for NP-hard graph completion (that is, edge addition) cases while we show that there is no hope to achieve analogous results for the corresponding vertex or edge deletion versions. Our algorithms are based on transforming graph completion problems into efficiently solvable number problems and exploiting f-factor computations for translating the results back into the graph setting. Our core observation is that we encounter a win-win situation: either the number of edge additions is small or the problem is polynomial-time solvable. This approach helps in answering an open question by Mathieson and Szeider [JCSS 2012] concerning the polynomial kernelizability of Degree Constraint Edge Addition and leads to a general method of approaching polynomial-time preprocessing for a wider class of degree sequence completion problems.

The Complexity of Degree Anonymization by Vertex Addition

Motivated by applications in privacy-preserving data publishing, we study the problem of making an undirected graph k-anonymous by adding few vertices (together with incident edges). That is, after adding these “dummy vertices”, for every vertex degree d in the resulting graph, there shall be at least k vertices with degree d. We explore three variants of vertex addition (justified by real-world considerations) and study their (parameterized) computational complexity. We derive mostly (worst-case) intractability results, even for very restricted cases (including trees or bounded-degree graphs) but also obtain a few encouraging fixed-parameter tractability results.

The effect of homogeneity on the computational complexity of combinatorial data anonymization

A matrix M is said to be k-anonymous if for each row r in M there are at least k − 1 other rows in M which are identical to r. The NP-hard k-Anonymity problem asks, given an n × m-matrix M over a fixed alphabet and an integer s > 0, whether M can be made k-anonymous by suppressing (blanking out) at most s entries. Complementing previous work, we introduce two new “data-driven” parameterizations for k-Anonymity—the number tin of different input rows and the number tout of different output rows—both modeling aspects of data homogeneity. We show that k-Anonymity is fixed-parameter tractable for the parameter tin, and that it is NP-hard even for tout = 2 and alphabet size four. Notably, our fixed-parameter tractability result implies that k-Anonymity can be solved in linear time when tin is a constant. Our computational hardness results also extend to the related privacy problems p-Sensitivity and ℓ-Diversity, while our fixed-parameter tractability results extend to p-Sensitivity and the usage of domain generalization hierarchies, where the entries are replaced by more general data instead of being completely suppressed.

A refined complexity analysis of degree anonymization in graphs

Motivated by a strongly growing interest in graph anonymization in the data mining and databases communities, we study the NP-hard problem of making a graph k-anonymous by adding as few edges as possible. Herein, a graph is k-anonymous if for every vertex in the graph there are at least k−1 other vertices of the same degree. Our algorithmic results shed light on the performance quality of a popular heuristic due to Liu and Terzi [ACM SIGMOD 2008]; in particular, we show that the heuristic provides optimal solutions in case that many edges need to be added. Based on this, we develop a polynomial-time data reduction, yielding a polynomial-size problem kernel for the problem parameterized by the maximum vertex degree. This result is in a sense tight since we also show that the problem is already NP-hard for H-index three, implying NP-hardness for smaller parameters such as average degree and degeneracy.

Parameterized Inapproximability of Degree Anonymization

The Degree Anonymity problem arises in the context of combinatorial graph anonymization. It asks, given a graph \(G\) and two integers \(k\) and \(s\), whether \(G\) can be made k-anonymous with at most \(s\) modifications. Here, a graph is k-anonymous if the graph contains for every vertex at least \(k-1\) other vertices of the same degree. Complementing recent investigations on its computational complexity, we show that this problem is very hard when studied from the viewpoints of approximation as well as parameterized approximation. In particular, for the optimization variant where one wants to minimize the number of either edge or vertex deletions there is no factor-\(n^{1-\varepsilon }\) approximation running in polynomial time unless P = NP, for any constant \(0 < \varepsilon \le 1\). For the variant where one wants to maximize \(k\) and the number \(s\) of either edge or vertex deletions is given, there is no factor-\(n^{{1}/{2}-\varepsilon }\) approximation running in time \(f(s) \cdot n^{O(1)}\) unless W[1] = FPT, for any constant \(0 < \varepsilon \le {1}/{2}\) and any function \(f\). On the positive side, we classify the general decision version as fixed-parameter tractable with respect to the combined parameter solution size \(s\) and maximum degree.

Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

Given an undirected graph G = (V,E) and an integer ` ≥ 1, the NPhard 2-Club problem asks for a vertex set S ⊆ V of size at least ` such that the subgraph induced by S has diameter at most two. In this work, we extend previous parameterized complexity studies for 2-Club. On the positive side, we give polynomial-size problem kernels for the parameters feedback edge set size of G and size of a cluster editing set of G and present a direct combinatorial algorithm for the parameter treewidth of G. On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit a polynomial-size problem kernel with respect to the size of a vertex cover of G. Next, we show that, under the strong exponential time hypothesis, a previous O(2|V |−` · |V ||E|)-time search tree algorithm [Schäfer et al., Optim. Lett. 2012] cannot be improved and that, unless NP ⊆ coNP/poly, there is no polynomial-size problem kernel for the dual parameter |V | − `. Finally, we show that, in spite of this lower bound, the search tree algorithm for the dual parameter |V | − ` can be tuned into an efficient exact algorithm for 2-Club that outperforms previous implementations. Submitted: July 2013 Reviewed: May 2014 Revised: August 2014 Accepted: February 2015 Final: February 2015 Published: March 2015 Article type: Regular paper Communicated by: P. Mutzel E-mail addresses: sepp.hartung@tu-berlin.de (Sepp Hartung) christian.komusiewicz@tu-berlin.de (Christian Komusiewicz) andre.nichterlein@tu-berlin.de (André Nichterlein) 156 Hartung et al. Algorithms and Experiments for Finding 2-Clubs

Parameterized Complexity of DAG Partitioning

The goal of tracking the origin of short, distinctive phrases (memes) that propagate through the web in reaction to current events has been formalized as DAG Partitioning: given a directed acyclic graph, delete edges of minimum weight such that each resulting connected component of the underlying undirected graph contains only one sink. Motivated by NP-hardness and hardness of approximation results, we consider the parameterized complexity of this problem. We show that it can be solved in O(2 k ·n 2) time, where k is the number of edge deletions, proving fixed-parameter tractability for parameter k. We then show that unless the Exponential Time Hypothesis (ETH) fails, this cannot be improved to 2 o(k) ·n O(1); further, DAG Partitioning does not have a polynomial kernel unless NP ⊆ coNP/poly. Finally, given a tree decomposition of width w, we show how to solve DAG Partitioning in \(2^{O(w^2)}\cdot n\) time, improving a known algorithm for the parameter pathwidth.