arXiv subjects

Eva Deltl

Publications and source records attributed to Eva Deltl.

7 recordsLinked to original sources

arXiv2026

Reaching Fairness by Reallocating Goods

Fair allocation of indivisible goods has largely been studied under the assumption that no prior allocation exists. Motivated by practical settings with pre-existing (and possibly unfair) allocations, we study how to achieve fairness through limited reallocations. Building on recent work on reformability/reallocations, we consider three fairness notions---envy-freeness (EF), envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX)---and optimize the number of goods reallocated. We analyze both the classical and parameterized complexity of these problems, providing a comprehensive analysis across multiple fairness notions.

cs.GT↗

arXiv2026

Optimally Selecting Representative Agents from a Metric Space

This paper studies the problem of proportionally fair clustering, where the goal is to select $k$ ``centers'' from a metric space that fairly represent a set of agents who also lie in the metric space. Specifically, we focus on finding a clustering satisfying a fairness property known as the Droop core. In the practical special case in which the set of feasible center locations contains every agent location, the previous best-known result guaranteed a $(1 + \sqrt{2})$-approximation of the Droop core, while the best-known lower bound was $2$. In this paper, we show that this lower bound is tight and that a clustering in the $2$-Droop core always exists. Further, we show that such a clustering can be achieved by only selecting centers from locations in the metric space where an agent resides. We establish this using Scarf's theorem guaranteeing a nonempty core for balanced non-transferable utility games. This result has several interesting corollaries. Most notably, it resolves the $\beta$-plurality problem of Aronov et al. [2021] for general metric spaces. The main result of this paper was generated by $\mathtt{ChatGPT}$-$\mathtt{5.6}$-$\mathtt{Sol}$ through a series of interactions with the authors. The authors of this paper verified the generated proof and rewrote it for clarity.

cs.GT↗

arXiv2026

The Influence of Agent Models on the Complexity of Bus Routing

In bus routing, the task is to plan a bus route in a network with several agents, each of whom wants to travel from a starting point to a destination. A bus route should account for several factors, including agents' cost for reaching the bus stops, their travel time, or the energy consumption of the buses. We study the complexity of several variants of this problem, focusing on how the objective function and the models for agents' walking costs influence the problem complexity. After observing that even the simplest agent cost model leads to hardness on general networks, we consider networks with tree structure. Our main findings are as follows. First, allowing agent-specific cost models leads to hardness even on extremely limited trees such as stars. Second, consistent agent models (where agents differ only in their starting points and destinations) make the problem easier in some cases. Finally, allowing agents to choose between using the bus and walking directly can make the problem considerably harder. Most of our hardness results show not only classical NP-hardness but also parameterized intractability for the natural parameter $k$, the number of bus stops.

cs.CC↗

arXiv2026

Improved Lower Bounds for Proportionally Fair Clustering

We study proportionally fair clustering, where a set of $k$ centers must be chosen from a metric space to represent $n$ agents, and no sufficiently large group of agents should be collectively underrepresented. One of the central notions of fairness in this setting is the $\alpha$-core. The existence of clusterings in the $(1+\sqrt{2})$-core was established by Chen et al. [2019], who also showed instances where the $\alpha$-core is empty for every $\alpha < 2$. Closing this gap has remained an open problem for seven years. We make progress from the lower-bound side by providing an instance whose $\alpha$-core is empty for every $\alpha < 2.1508$. Our techniques rely on establishing connections between variants of the core, namely the Hare core and the Droop core; reducing the search for optimal empty-core instances to a highly structured family of clustering instances; and using a Mixed Integer Linear Program (MILP) to search for optimal lower-bound instances within this reduced space. Using this framework, we also determine tight bounds for Droop quota clustering instances with a small number of possible candidate centers and a single center to be selected. For each number of centers $m \in \{3,4,5,6\}$, we give the exact threshold $\alpha_m^*$ such that an $\alpha_m^*$-core clustering always exists, while for every $\alpha < \alpha_m^*$ there is an instance with $m$ centers whose $\alpha$-core is empty. Although these values were originally found through computer-aided search, we also provide direct proofs that do not rely on MILP certificates.

cs.GT↗

arXiv2026

Minimizing the Cost of EFx Allocations

Ensuring fairness while limiting costs, such as transportation or storage, is an important challenge in resource allocation, yet most work has focused on cost minimization without fairness or fairness without explicit cost considerations. We introduce and formally define the minCost-EFx Allocation problem, where the objective is to compute an allocation that is envy-free up to any item (EFx) and has minimum cost. We investigate the algorithmic complexity of this problem, proving that it is NP-hard already with two agents. On the positive side, we show that the problem admits a polynomial kernel with respect to the number of items, implying that a core source of intractability lies in the number of items. Building on this, we identify parameter-restricted settings that are tractable, including cases with bounded valuations and a constant number of agents, or a limited number of item types under restricted cost models. Finally, we turn to cost approximation, proving that for any $\rho>1$ the problem is not $\rho$-approximable in polynomial time (unless $P=NP$), while also identifying restricted cost models where costs are agent-specific and independent of the actual items received, which admit better approximation guarantees.

cs.GT↗

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 $\alpha$-core for any $\alpha<2^{\frac{1}{5}}\sim 1.148$. 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

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↗