Modification-Fair Cluster Editing

The classic Cluster Editing problem (also known as Correlation Clustering) asks to transform a given graph into a disjoint union of cliques (clusters) by a small number of edge modifications. When applied to vertex-colored graphs (the colors representing subgroups), standard algorithms for the NP-hard Cluster Editing problem may yield solutions that are biased towards subgroups of data (e.g., demographic groups), measured in the number of modifications incident to the members of the subgroups. We propose a modification fairness constraint which ensures that the number of edits incident to each subgroup is proportional to its size. To start with, we study Modification-Fair Cluster Editing for graphs with two vertex colors. We show that the problem is NP-hard even if one may only insert edges within a subgroup; note that in the classic "non-fair" setting, this case is trivially polynomial-time solvable. However, in the more general editing form, the modification-fair variant remains fixed-parameter tractable with respect to the number of edge edits. We complement these and further theoretical results with an empirical analysis of our model on real-world social networks where we find that the price of modification-fairness is surprisingly low, that is, the cost of optimal modification-fair differs from the cost of optimal "non-fair" solutions only by a small percentage.

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present several results on the parameterized complexity of training two-layer ReLU networks with respect to various loss functions. After a brief discussion of other parameters, we focus on analyzing the influence of the dimension $d$ of the training data on the computational complexity. We provide running time lower bounds in terms of W[1]-hardness for parameter $d$ and prove that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis). In comparison with previous work, our results hold for a broad(er) range of loss functions, including $\ell^p$-loss for all $p\in[0,\infty]$. In particular, we extend a known polynomial-time algorithm for constant $d$ and convex loss functions to a more general class of loss functions, matching our running time lower bounds also in these cases.

Comparing Temporal Graphs Using Dynamic Time Warping

The connections within many real-world networks change over time. Thus, there has been a recent boom in studying temporal graphs. Recognizing patterns in temporal graphs requires a similarity measure to compare different temporal graphs. To this end, we initiate the study of dynamic time warping (an established concept for mining time series data) on temporal graphs. We propose the dynamic temporal graph warping distance (dtgw) to determine the (dis-)similarity of two temporal graphs. Our novel measure is flexible and can be applied in various application domains. We show that computing the dtgw-distance is a challenging (NP-hard) optimization problem and identify some polynomial-time solvable special cases. Moreover, we develop a quadratic programming formulation and an efficient heuristic. Preliminary experiments indicate that the heuristic performs very well and that our concept yields meaningful results on real-world instances.

Complexity of Shift Bribery in Committee Elections

Given an election, a preferred candidate p, and a budget, the SHIFT BRIBERY problem asks whether p can win the election after shifting p higher in some voters' preference orders. Of course, shifting comes at a price (depending on the voter and on the extent of the shift) and one must not exceed the given budget. We study the (parameterized) computational complexity of S HIFT BRIBERY for multiwinner voting rules where winning the election means to be part of some winning committee. We focus on the well-established SNTV, Bloc, k-Borda, and Chamberlin-Courant rules, as well as on approximate variants of the Chamberlin-Courant rule, since the original rule is NP-hard to compute. We show that SHIFT BRIBERY tends to be harder in the multiwinner setting than in the single-winner one by showing settings where SHIFT BRIBERY is easy in the single-winner cases, but is hard (and hard to approximate) in the multiwinner ones. Moreover, we show that the non-monotonicity of those rules which are based on approximation algorithms for the Chamberlin-Courant rule sometimes affects the complexity of SHIFT BRIBERY.

Exact Mean Computation in Dynamic Time Warping Spaces

Dynamic time warping constitutes a major tool for analyzing time series. In particular, computing a mean series of a given sample of series in dynamic time warping spaces (by minimizing the Fr\'echet function) is a challenging computational problem, so far solved by several heuristic and inexact strategies. We spot some inaccuracies in the literature on exact mean computation in dynamic time warping spaces. Our contributions comprise an exact dynamic program computing a mean (useful for benchmarking and evaluating known heuristics). Based on this dynamic program, we empirically study properties like uniqueness and length of a mean. Moreover, experimental evaluations reveal substantial deficits of state-of-the-art heuristics in terms of their output quality. We also give an exact polynomial-time algorithm for the special case of binary time series.

Elections with Few Voters: Candidate Control Can Be Easy

We study the computational complexity of candidate control in elections with few voters, that is, we consider the parameterized complexity of candidate control in elections with respect to the number of voters as a parameter. We consider both the standard scenario of adding and deleting candidates, where one asks whether a given candidate can become a winner (or, in the destructive case, can be precluded from winning) by adding or deleting few candidates, as well as a combinatorial scenario where adding/deleting a candidate automatically means adding or deleting a whole group of candidates. Considering several fundamental voting rules, our results show that the parameterized complexity of candidate control, with the number of voters as the parameter, is much more varied than in the setting with many voters.