Leon Kellerhals

Leon Kellerhals, Jannik Peters

We study the proportional clustering problem of Chen et al. [ICML'19] and relate it to the area of multiwinner voting in computational social choice. We show that any clustering satisfying a weak proportionality notion of Brill and Peters [EC'23] simultaneously obtains the best known approximations to the proportional fairness notion of Chen et al. [ICML'19], but also to individual fairness [Jung et al., FORC'20] and the "core" [Li et al. ICML'21]. In fact, we show that any approximation to proportional fairness is also an approximation to individual fairness and vice versa. Finally, we also study stronger notions of proportional representation, in which deviations do not only happen to single, but multiple candidate centers, and show that stronger proportionality notions of Brill and Peters [EC'23] imply approximations to these stronger guarantees.

Leon Kellerhals, Mitja Krebs, André Nichterlein et al.

We study the following problem that is motivated by demand-aware network design: Given a tree~$G$, the task is to find a binary tree~$H$ on the same vertex set. The objective is to minimize the sum of distances in~$H$ between vertex pairs that are adjacent in~$G$. We present a linear-time factor-4 approximation for this problem.

Christian Wallisch, Andrea Benigni, Carsten Hartmann et al.

Electrical distribution networks are regional, medium- and low-voltage power grids connecting energy sources to individual households and businesses with given power demands. While these networks contain redundant power lines for reliability, they are typically operated in a radial (spanning tree) configuration by opening and closing switches on the lines. The challenge is to find a spanning tree that minimizes the sum of the resistive power losses: The power loss of a line $e$ is its resistance $r(e)$ times the squared current $f(e)^2$ flowing across the line. We study approximation algorithms for this problem, known as Distribution Network Reconfiguration (DNR). We give an $n$-approximation algorithm and, via a new NP-hardness for planar Balanced Connected Partition with a fixed number of parts, show that no $n^{1-\varepsilon}$-approximation is possible even on planar graphs unless P $=$ NP, for any $\varepsilon>0$. Since the approximation hardness holds only if there are many sources, we focus on $k$-DNR with $k$ sources; this is motivated by traditional distribution networks, where oftentimes $k = 1$. For $2$-DNR, we give an approximation lower bound of $Ω(\log^2 n)$ conditioned on P $\neq$ NP. For $1$-DNR, which is equivalent to finding an uncapacitated confluent flow minimizing the squared Euclidean norm, we prove APX-hardness and give an $\mathcal{O}(\sqrt{n})$-approximation for uniform line resistances, answering an open question by Gupta et al. [Math. Program. 2022].

Vincent Froese, Leon Kellerhals, Rolf Niedermeier

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 solutions differs from the cost of optimal "non-fair" solutions only by a small percentage.