Leon Kellerhals

IJCAI Conference 2025 Conference Paper

How to Resolve Envy by Adding Goods

• Matthias Bentert
• Robert Bredereck
• Eva Deltl
• Pallavi Jain
• Leon Kellerhals

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.

TCS Journal 2024 Journal Article

Approximating sparse quadratic programs

• Danny Hermelin
• Leon Kellerhals
• Rolf Niedermeier
• Rami Pugatch

AAAI Conference 2024 Conference Paper

Locally Rainbow Paths

• Till Fluschnik
• Leon Kellerhals
• Malte Renken

We introduce the algorithmic problem of finding a locally rainbow path of length l 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 leaset 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.

NeurIPS Conference 2024 Conference Paper

Proportional Fairness in Clustering: A Social Choice Perspective

• 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. , 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 imply approximations to these stronger guarantees.

AAAI Conference 2023 Conference Paper

Fair Short Paths in Vertex-Colored Graphs

• Matthias Bentert
• Leon Kellerhals
• Rolf Niedermeier

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.

AAAI Conference 2023 Conference Paper

Parameterized Algorithms for Colored Clustering

• Leon Kellerhals
• Tomohiro Koana
• Pascal Kunz
• Rolf Niedermeier

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 2ᴼ⁽ᵏ⁾·nᴼ⁽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 O(k⁵ᐟ²), 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 local feedback edge number.

AAAI Conference 2022 Conference Paper

Modification-Fair Cluster Editing

• 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 vertexcolored 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 CLUS- TER 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.

IJCAI Conference 2022 Conference Paper

Placing Green Bridges Optimally, with Habitats Inducing Cycles

• Maike Herkenrath
• Till Fluschnik
• Francesco Grothe
• Leon Kellerhals

Choosing the placement of wildlife crossings (i. e. , green bridges) to reconnect animal species' fragmented habitats is among the 17 goals towards sustainable development by the UN. We consider the following established model: Given a graph whose vertices represent the fragmented habitat areas and whose weighted edges represent possible green bridge locations, as well as the habitable vertex set for each species, find the cheapest set of edges such that each species' habitat is connected. We study this problem from a theoretical (algorithms and complexity) and an experimental perspective, while focusing on the case where habitats induce cycles. We prove that the NP-hardness persists in this case even if the graph structure is restricted. If the habitats additionally induce faces in plane graphs however, the problem becomes efficiently solvable. In our empirical evaluation we compare this algorithm as well as ILP formulations for more general variants and an approximation algorithm with another. Our evaluation underlines that each specialization is beneficial in terms of running time, whereas the approximation provides highly competitive solutions in practice.

IJCAI Conference 2022 Conference Paper

Single-Peaked Opinion Updates

• Robert Bredereck
• Anne-Marie George
• Jonas Israel
• Leon Kellerhals

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.

TCS Journal 2022 Journal Article

The structural complexity landscape of finding balance-fair shortest paths

• Matthias Bentert
• Leon Kellerhals
• Rolf Niedermeier

IJCAI Conference 2020 Conference Paper

Maximizing the Spread of an Opinion in Few Steps: Opinion Diffusion in Non-Binary Networks

• Robert Bredereck
• Lilian Jacobs
• Leon Kellerhals

We consider the setting of asynchronous opinion diffusion with majority threshold: given a social network with each agent assigned to one opinion, an agent will update its opinion if more than half of its neighbors agree on a different opinion. The stabilized final outcome highly depends on the sequence in which agents update their opinion. We are interested in optimistic sequences---sequences that maximize the spread of a chosen opinion. We complement known results for two opinions where optimistic sequences can be computed in time and length linear in the number of agents. We analyze upper and lower bounds on the length of optimistic sequences, showing quadratic bounds in the general and linear bounds in the acyclic case. Moreover, we show that in networks with more than two opinions determining a spread-maximizing sequence becomes intractable; surprisingly, already with three opinions the intractability results hold in highly restricted cases, e. g. , when each agent has at most three neighbors, when looking for a short sequence, or when we aim for approximate solutions.