Manuel Sorge

Exact combinatorial algorithms and experiments for finding maximum k-plexes

We propose new practical algorithms to find maximum-cardinality k-plexes in graphs. A k-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most k vertices in the k-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality k-plexes is NP-hard. Complementing previous work, we develop exact combinatorial algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.

Algorithms and Experiments for Clique Relaxations--Finding Maximum s-Plexes

We propose new practical algorithms to find degree-relaxed variants of cliques called s -plexes. An s -plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most s vertices in the s -plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality s -plexes is NP-hard. Complementing previous work, we develop combinatorial, exact algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.

A Parameterized Complexity Analysis of Combinatorial Feature Selection Problems

We examine the algorithmic tractability of NP-hard combinatorial feature selection problems in terms of parameterized complexity theory. In combinatorial feature selection, one seeks to discard dimensions from high-dimensional data such that the resulting instances fulfill a desired property. In parameterized complexity analysis, one seeks to identify relevant problem-specific quantities and tries to determine their influence on the computational complexity of the considered problem. In this paper, for various combinatorial feature selection problems, we identify parameterizations and reveal to what extent these govern computational complexity. We provide tractability as well as intractability results; for example, we show that the Distinct Vectors problem on binary points is polynomial-time solvable if each pair of points differs in at most three dimensions, whereas it is NP-hard otherwise.

H-index manipulation by merging articles: models, theory, and experiments

Google Scholar allows merging multiple article versions into one. This merging affects the H-index computed by Google Scholar. We analyze the parameterized complexity of maximizing the H-index using article merges. Herein, multiple possible measures for computing the citation count of a merged article are considered. Among others, for the measure used by Google Scholar, we give an algorithm that maximizes the H-index in linear time if there is only a constant number of versions of the same article. In contrast, if we are allowed to merge arbitrary articles, then already increasing the H-index by one is NP-hard.An authors profile on Google Scholar consists of indexed articles and associated data, such as the number of citations and the H-index. The author is allowed to merge articles; this may affect the H-index. We analyze the (parameterized) computational complexity of maximizing the H-index using article merges. Herein, to model realistic manipulation scenarios, we define a compatibility graph whose edges correspond to plausible merges. Moreover, we consider several different measures for computing the citation count of a merged article. For the measure used by Google Scholar, we give an algorithm that maximizes the H-index in linear time if the compatibility graph has constant-size connected components. In contrast, if we allow to merge arbitrary articles (that is, for compatibility graphs that are cliques), then already increasing the H-index by one is NP-hard. Experiments on Google Scholar profiles of AI researchers show that the H-index can be manipulated substantially only if one merges articles with highly dissimilar titles.

Finding Highly Connected Subgraphs

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 first observe that this 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), and expose their influence on the complexity of this problem. 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 parameter value in relevant instances.

An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems

We investigate the computational complexity of the Densestk-Subgraph problem, where the input is an undirected graph G=(V,E) and one wants to find a subgraph on exactly k vertices with the maximum number of edges. We extend previous work on Densestk-Subgraph by studying its parameterized complexity for parameters describing the sparseness of the input graph and for parameters related to the solution size k.On the positive side, we show that, when fixing some constant minimum density of the sought subgraph, Densestk-Subgraph becomes fixed-parameter tractable with respect to either of the parameters maximum degree of G and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for Densestk-Subgraph with respect to the combined parameter degeneracy of G and |V|k.On the negative side, we find that Densestk-Subgraph is W[1]-hard with respect to the combined parameter solution size k and degeneracy of G. We furthermore strengthen a previous hardness result for Densestk-Subgraph (Cai, 2008) by showing that for every fixed , 0<<1, the problem of deciding whether G contains a subgraph of density at least is W[1]-hard with respect to the parameter |V|k.Our positive results are obtained by an algorithmic framework that can be applied to a wide range of Fixed-Cardinality Optimization problems.

Enumerating maximal cliques in temporal graphs

Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this are temporal graphs. We focus on enumerating Δ-cliques, an extension of the concept of cliques to temporal graphs: for a given time period Δ, a Δ-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every Δ time steps within the time interval. Viard, Latapy, and Magnien [ASONAM 2015] proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt to the temporal setting the Bron-Kerbosch algorithm - an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs. We obtain encouraging results both in theory (concerning worst-case time analysis based on the parameter “Δ-slice degeneracy” of the underlying graph) as well as in practice with experiments on real-world data. The latter culminates in a significant improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien (typically two orders of magnitude).

On the Parameterized Complexity of Computing Balanced Partitions in Graphs

A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the VertexBisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that VertexBisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of nO(d). We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.

On the Parameterized Complexity of Computing Graph Bisections

The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published that consider the parameterized complexity of this problem.

Assessing the Computational Complexity of Multi-layer Subgraph Detection

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability.