Sepp Hartung

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”.

Linear-Time computation of a linear problem kernel for dominating set on planar graphs

We present a linear-time kernelization algorithm that transforms a given planar graph G with domination number γ(G) into a planar graph G′ of size O(γ(G)) with γ(G)=γ(G′). In addition, a minimum dominating set for G can be inferred from a minimum dominating set for G′. In terms of parameterized algorithmics, this implies a linear-size problem kernel for the NP-hard Dominating Set problem on planar graphs, where the kernelization takes linear time. This improves on previous kernelization algorithms that provide linear-size kernels in cubic time.

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.

A multivariate complexity analysis of lobbying in multiple referenda

We extend work by Christian et al. [Review of Economic Design 2007] on lobbying in multiple referenda by first providing a more fine-grained analysis of the computational complexity of the NP-complete LOBBYING problem. Herein, given a binary matrix, the columns represent issues to vote on and the rows correspond to voters making a binary vote on each issue. An issue is approved if a majority of votes has a 1 in the corresponding column. The goal is to get all issues approved by modifying a minimum number of rows to all-1-rows. In our multivariate complexity analysis, we present a more holistic view on the nature of the computational complexity of LOBBYING, providing both (parameterized) tractability and intractability results, depending on various problem parameterizations to be adopted. Moreover, we show non-existence results concerning efficient and effective preprocessing for LOBBYING and introduce natural variants such as RESTRICTED LOBBYING and PARTIAL LOBBYING.

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.

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.

Programming by Optimisation Meets Parameterised Algorithmics: A Case Study for Cluster Editing

Inspired by methods and theoretical results from parameterised algorithmics, we improve the state of the art in solving Cluster Editing, a prominent NP-hard clustering problem with applications in computational biology and beyond. In particular, we demonstrate that an extension of a certain preprocessing algorithm, called the \((k+1)\)-data reduction rule in parameterised algorithmics, embedded in a sophisticated branch-&-bound algorithm, improves over the performance of existing algorithms based on Integer Linear Programming (ILP) and branch-&-bound. Furthermore, our version of the \((k+1)\)-rule outperforms the theoretically most effective preprocessing algorithm, which yields a 2k-vertex kernel. Notably, this 2k-vertex kernel is analysed empirically for the first time here. Our new algorithm was developed by integrating Programming by Optimisation into the classical algorithm engineering cycle – an approach which we expect to be successful in many other contexts.

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

The Parameterized Complexity of Local Search for TSP, More Refined

We extend previous work on the parameterized complexity of local search for the Traveling Salesperson Problem (TSP). So far, its parameterized complexity has been investigated with respect to the distance measures (defining the local search area) “Edge Exchange” and “Max-Shift”. We perform studies with respect to the distance measures “Swap” and “r-Swap”, “Reversal” and “r-Reversal”, and “Edit”, achieving both fixed-parameter tractability and W[1]-hardness results. In particular, from the parameterized reduction showing W[1]-hardness we infer running time lower bounds (based on the exponential time hypothesis) for all corresponding distance measures. Moreover, we provide non-existence results for polynomial-size problem kernels and we show that some in general W[1]-hard problems turn fixed-parameter tractable when restricted to planar graphs.

The Complexity of Degree Anonymization by Graph Contractions

We study the computational complexity of \(k\)-anonymizing a given graph by as few graph contractions as possible. A graph is said to be \(k\)-anonymous if for every vertex in it, there are at least \(k-1\) other vertices with exactly the same degree. The general degree anonymization problem is motivated by applications in privacy-preserving data publishing, and was studied to some extent for various graph operations (most notable operations being edge addition, edge deletion, vertex addition, and vertex deletion). We complement this line of research by studying several variants of graph contractions, which are operations of interest, for example, in the contexts of social networks and clustering algorithms. We show that the problem of degree anonymization by graph contractions is \({\mathsf {NP}}\)-hard even for some very restricted inputs, and identify some fixed-parameter tractable cases.