arXiv subjects

Leon Kellerhals

Publications and source records attributed to Leon Kellerhals.

At least 19 recordsLinked to original sources

arXiv2026

Designing Caterpillars for Graphs: Approximation and Hardness

The classical Minimum Linear Arrangement (MLA) problem has been studied extensively. It is known to be NP-hard and it admits an $O(\sqrt{\log n}\log\log n)$-approximation [Feige and Lee, IPL, 2007]. MLA can be defined as follows as design problem: Given a graph $G$ with vertex set $V(G)$, design a path $H$ on the same vertex set that minimizes the linear arrangement cost $\sum_{uv\in E(G)}\textrm{dist}_H(u,v)$, where $\textrm{dist}_H(u,v)$ indicates the distance of $u$ and $v$ in $H$. We initiate the study of the generalization in which $H$ is allowed to be a caterpillar graph of maximum degree at most $Δ$. Caterpillars are the simplest generalization of paths, having pathwidth one and interpolating between paths and stars via the degree parameter $Δ$. We give an algorithm that lifts any $α$-approximation for MLA to an $(α+3-2/(Δ-1))$-approximation for our problem, thus obtaining an $O(\sqrt{\log n}\log\log n)$-approximation for our more general problem as well. Moreover, we derive a $4$-approximation whenever MLA is polynomial-time solvable, in particular, for trees. Complementing these results, we prove NP-hardness for every constant $Δ\geq 2$, and, in stark contrast to MLA, show it remains NP-hard on trees when $Δ$ is part of the input.

cs.DS↗

arXiv2026

Approximability of Electrical Distribution Network Reconfiguration for General Graphs

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].

cs.DS↗

arXiv2026

Designing Approximate Binary Trees for Trees

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.

cs.DS↗

arXiv2025

The Core in Max-Loss Non-Centroid Clustering Can Be Empty

We study core stability in non-centroid clustering under the max-loss objective, where each agent's loss is the maximum distance to other members of their cluster. We prove that for all $k\geq 3$ there exist metric instances with $n\ge 9$ agents, with $n$ divisible by $k$, for which no clustering lies in the $α$-core for any $α<2^{\frac{1}{5}}\sim 1.148$. The bound is tight for our construction. Using a computer-aided proof, we also identify a two-dimensional Euclidean point set whose associated lower bound is slightly smaller than that of our general construction. This is, to our knowledge, the first impossibility result showing that the core can be empty in non-centroid clustering under the max-loss objective.

cs.LG↗

arXiv2025

The Harmonic Policy for Online Buffer Sharing is (2 + ln n)-Competitive: A Simple Proof

The problem of online buffer sharing is expressed as follows. A switch with $n$ output ports receives a stream of incoming packets. When an incoming packet is accepted by the switch, it is stored in a shared buffer of capacity $B$ common to all packets and awaits its transmission through its corresponding output port determined by its destination. Each output port transmits one packet per time unit. The problem is to find an algorithm for the switch to accept or reject a packet upon its arrival in order to maximize the total number of transmitted packets. Building on the work of Kesselman et al. (STOC 2001) on split buffer sharing, Kesselman and Mansour (TCS 2004) considered the problem of online buffer sharing which models most deployed internet switches. In their work, they presented the Harmonic policy and proved that it is $(2 + \ln n)$-competitive, which is the best known competitive ratio for this problem. The Harmonic policy unfortunately saw less practical relevance as it performs $n$ threshold checks per packets which is deemed costly in practice, especially on network switches processing multiple terabits of packets per second. While the Harmonic policy is elegant, the original proof is also rather complex and involves a lengthy matching routine along with multiple intermediary results. This note presents a simplified Harmonic policy, both in terms of implementation and proof. First, we show that the Harmonic policy can be implemented with a constant number of threshold checks per packet, matching the widely deployed \emph{Dynamic Threshold} policy. Second, we present a simple proof that shows the Harmonic policy is $(2 + \ln n)$-competitive. In contrast to the original proof, the current proof is direct and relies on a 3-partitioning of the packets.

cs.DS↗

arXiv2025

Placing Green Bridges Optimally, with Close-Range Habitats in Sparse Graphs

We study a network design problem motivated by the challenge of placing wildlife crossings to reconnect fragmented habitats of animal species, which is among the 17 goals towards sustainable development by the UN: Given a graph, whose vertices represent the fragmented habitat areas and whose edges represent possible green bridge locations (with costs), and the habitable vertex set for each species' habitat, the goal is to find the cheapest set of edges such that each species' habitat is sufficiently connected. We focus on the established variant where a habitat is considered sufficiently connected if it has diameter two in the solution and study its complexity in cases justified by our setting namely small habitat sizes on planar graphs and graphs of small maximum degree $Δ$. We provide efficient algorithms and NP-hardness results for different values of $Δ$ and maximum habitat sizes on general and planar graphs.

cs.DS↗

arXiv2025

How to Resolve Envy by Adding Goods

We consider the problem of resolving the envy of a given initial allocation by adding elements from a pool of goods. We give a characterization of the instances where envy can be resolved by adding an arbitrary number of copies of the items in the pool. From this characterization, we derive a polynomial-time algorithm returning a respective solution if it exists. If the number of copies or the total number of added items are bounded, the problem becomes computationally intractable even in various restricted cases. We perform a parameterized complexity analysis, focusing on the number of agents and the pool size as parameters. Notably, although not every instance admits an envy-free solution, our approach allows us to efficiently determine, in polynomial time, whether a solution exists-an aspect that is both theoretically interesting and far from trivial.

cs.GT↗

arXiv2025

Proportional Clustering, the $β$-Plurality Problem, and Metric Distortion

We show that the proportional clustering problem using the Droop quota for $k = 1$ is equivalent to the $β$-plurality problem. We also show that the Plurality Veto rule can be used to select ($\sqrt{5} - 2$)-plurality points using only ordinal information about the metric space and resolve an open question of Kalayci et al. (AAAI 2024) by proving that $(2+\sqrt{5})$-proportionally fair clusterings can be found using purely ordinal information.

cs.GT↗

arXiv2024

Structural Parameterizations of the Biclique-Free Vertex Deletion Problem

In this work, we study the Biclique-Free Vertex Deletion problem: Given a graph $G$ and integers $k$ and $i \le j$, find a set of at most $k$ vertices that intersects every (not necessarily induced) biclique $K_{i, j}$ in $G$. This is a natural generalization of the Bounded-Degree Deletion problem, wherein one asks whether there is a set of at most $k$ vertices whose deletion results in a graph of a given maximum degree $r$. The two problems coincide when $i = 1$ and $j = r + 1$. We show that Biclique-Free Vertex Deletion is fixed-parameter tractable with respect to $k + d$ for the degeneracy $d$ by developing a $2^{O(d k^2)} \cdot n^{O(1)}$-time algorithm. We also show that it can be solved in $2^{O(f k)} \cdot n^{O(1)}$ time for the feedback vertex number $f$ when $i \ge 2$. In contrast, we find that it is W[1]-hard for the treedepth for any integer $i \ge 1$. Finally, we show that Biclique-Free Vertex Deletion has a polynomial kernel for every $i \ge 1$ when parameterized by the feedback edge number. Previously, for this parameter, its fixed-parameter tractability for $i = 1$ was known (Betzler et al., 2012) but the existence of polynomial kernel was open.

cs.DS↗

arXiv2024

The Structural Complexity Landscape of Finding Balance-Fair Shortest Paths

We study the parameterized complexity of finding shortest s-t-paths with an additional fairness requirement. The task is to compute a shortest path in a vertex-colored graph where each color appears (roughly) equally often in the solution. We provide a complete picture of the parameterized complexity landscape of the problem with respect to structural parameters by showing a tetrachotomy including polynomial kernels, fixed-parameter tractability, XP-time algorithms (and W[1]-hardness), and para-NP-hardness.

cs.DS↗

arXiv2024

Proportional Fairness in Clustering: A Social Choice Perspective

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.

cs.LG↗

arXiv2024

Placing Green Bridges Optimally, with a Multivariate Analysis

We study the problem of placing wildlife crossings, such as green bridges, over human-made obstacles to challenge habitat fragmentation. The main task herein is, given a graph describing habitats or routes of wildlife animals and possibilities of building green bridges, to find a low-cost placement of green bridges that connects the habitats. We develop different problem models for this task and study them from a computational complexity and parameterized algorithmics perspective.

cs.CC↗

arXiv2024

Locally Rainbow Paths

We introduce the algorithmic problem of finding a locally rainbow path of length $\ell$ connecting two distinguished vertices $s$ and $t$ in a vertex-colored directed graph. Herein, a path is locally rainbow if between any two visits of equally colored vertices, the path traverses consecutively at least $r$ differently colored vertices. This problem generalizes the well-known problem of finding a rainbow path. It finds natural applications whenever there are different types of resources that must be protected from overuse, such as crop sequence optimization or production process scheduling. We show that the problem is computationally intractable even if $r=2$ or if one looks for a locally rainbow among the shortest paths. On the positive side, if one looks for a path that takes only a short detour (i.e., it is slightly longer than the shortest path) and if $r$ is small, the problem can be solved efficiently. Indeed, the running time of the respective algorithm is near-optimal unless the ETH fails.

cs.DS↗

arXiv2024

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

cs.DS↗

arXiv2023

Fair Short Paths in Vertex-Colored Graphs

The computation of short paths in graphs with arc lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. For the setting where each vertex is assigned to a group (color), we provide a framework to model multiple natural fairness aspects. We seek to find short paths in which the number of occurrences of each color is within some given lower and upper bounds. Among other results, we prove the introduced problems to be computationally intractable (NP-hard and parameterized hard with respect to the number of colors) even in very restricted settings (such as each color should appear with exactly the same frequency), while also presenting an encouraging algorithmic result ("fixed-parameter tractability") related to the length of the sought solution path for the general problem.

cs.DS↗

arXiv2023

Parameterized Algorithms for Colored Clustering

In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a $2^{\mathcal O(k)} \cdot n^{\mathcal O(1)}$-time algorithm where $k$ is the number of edges to be selected and $n$ the number of vertices. We also prove the existence of a problem kernel of size $\mathcal O(k^{5/2} )$, resolving an open problem posed in the literature. We consider parameters that are smaller than $k$, the number of edges to be selected, and $r$, the number of edges that can be deleted. Such smaller parameters are obtained by considering the difference between $k$ or $r$ and some lower bound on these values. We give both algorithms and lower bounds for Colored Clustering with such parameterizations. Finally, we settle the parameterized complexity of Colored Clustering with respect to structural graph parameters by showing that it is $W[1]$-hard with respect to both vertex cover number and tree-cut width, but fixed-parameter tractable with respect to slim tree-cut width.

cs.DS↗

arXiv2022

Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.

cs.DS↗

arXiv2022

Single-Peaked Opinion Updates

We consider opinion diffusion for undirected networks with sequential updates when the opinions of the agents are single-peaked preference rankings. Our starting point is the study of preserving single-peakedness. We identify voting rules that, when given a single-peaked profile, output at least one ranking that is single peaked w.r.t. a single-peaked axis of the input. For such voting rules we show convergence to a stable state of the diffusion process that uses the voting rule as the agents' update rule. Further, we establish an efficient algorithm that maximises the spread of extreme opinions.

cs.GT↗