arXiv subjects

Zsuzsanna Jankó

Publications and source records attributed to Zsuzsanna Jankó.

7 recordsLinked to original sources

arXiv2026

Well-order a flame

An $r$-rooted (possibly infinite) digraph $ D=(V,E) $ is a flame if for every $ v\in V\setminus \{ r \} $ there exists a set of edge-disjoint paths from $r$ to $v$ in $D$ that covers all ingoing edges of $ v $. Flames were first studied by Lovász in his investigation of edge-minimal subgraphs of a rooted digraph that preserve all the local edge-connectivities from the root. He showed that these subgraphs are always flames. Szeszlér later proved a common generalisation of Lovász' result and Edmonds' disjoint arborescence theorem. In this paper we focus on infinite flames and prove the following constructive characterisation. Every (possibly infinite) flame can be constructed transfinitely, starting from the empty edge set and adding a single edge at each step in such a way that every intermediate digraph is again a flame.

math.CO↗

arXiv2024

On Connected Strongly-Proportional Cake-Cutting

We investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents who may have different entitlements. We characterize the existence of a connected strongly-proportional allocation -- one in which every agent receives a contiguous piece worth strictly more than their proportional share. The characterization is supplemented with an algorithm that determines its existence using O(n * 2^n) queries. We devise a simpler characterization for agents with strictly positive valuations and with equal entitlements, and present an algorithm to determine the existence of such an allocation using O(n^2) queries. We provide matching lower bounds in the number of queries for both algorithms. When a connected strongly-proportional allocation exists, we show that it can also be computed using a similar number of queries. We also consider the problem of deciding the existence of a connected allocation of a cake in which each agent receives a piece worth a small fixed value more than their proportional share, and the problem of deciding the existence of a connected strongly-proportional allocation of a pie.

math.CO↗

arXiv2022

On generalisations of the Aharoni-Pouzet base exchange theorem

The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $ and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus X_i) \cup Y_i $ is a base for every $ i $. The special case where each $ X_i $ is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite dimensional vector spaces. Later he and Aharoni answered this conjecture affirmatively not just for vector spaces but for infinite matroids. We prove two generalisations of their result. On the one hand, we show that `being a singleton' can be relaxed to `being finite' and this is sharp in the sense the exclusion of infinite sets is really necessary. On the other hand, we prove that if $ B_0$ and $ B_1 $ are bases, then there is a bijection $ F $ between their finite subsets such that $ (B_0\setminus I) \cup F(I) $ is a base for every $ I$. In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.

math.CO↗

arXiv2022

Cutting a cake for infinitely many guests

Fair division with unequal shares is an intensively studied recourse allocation problem. For $ i\in [n] $, let $ μ_i $ be an atomless probability measure on the measurable space $(C,\mathcal{S}) $ and let $ t_i $ be positive numbers (entitlements) with $ \sum_{i=1}^{n}t_i=1 $. A fair division is a partition of $ C $ into sets $ S_i\in \mathcal{S} $ with $ μ_i(S_i)\geq t_i$ for every $ i\in [n] $. We introduce new algorithms to solve the fair division problem with irrational entitlements. They are based on the classical Last diminisher technique and we believe that they are simpler than the known methods. Then we show that a fair division always exists even for infinitely many players.

math.CO↗

arXiv2021

A quest for a fair schedule: The Young Physicists' Tournament

The Young Physicists Tournament is an established team-oriented scientific competition between high school students from 37 countries on 5 continents. The competition consists of scientific discussions called Fights. Three or four teams participate in each Fight, each of whom presents a problem while rotating the roles of Presenter, Opponent, Reviewer, and Observer among them. The rules of a few countries require that each team announce in advance 3 problems they will present at the national tournament. The task of the organizers is to choose the composition of Fights in such a way that each team presents each of its chosen problems exactly once and within a single Fight no problem is presented more than once. Besides formalizing these feasibility conditions, in this paper we formulate several additional fairness conditions for tournament schedules. We show that the fulfillment of some of them can be ensured by constructing suitable edge colorings in bipartite graphs. To find fair schedules, we propose integer linear programs and test them on real as well as randomly generated data.

cs.AI↗

arXiv2019

Complexity of Stability in Trading Networks

Efficient computability is an important property of solution concepts in matching markets. We consider the computational complexity of finding and verifying various solution concepts in trading networks-multi-sided matching markets with bilateral contracts-under the assumption of full substitutability of agents' preferences. It is known that outcomes that satisfy trail stability always exist and can be found in linear time. Here we consider a slightly stronger solution concept in which agents can simultaneously offer an upstream and a downstream contract. We show that deciding the existence of outcomes satisfying this solution concept is an NP-complete problem even in a special (flow network) case of our model. It follows that the existence of stable outcomes--immune to deviations by arbitrary sets of agents-is also an NP-hard problem in trading networks (and in flow networks). Finally, we show that even verifying whether a given outcome is stable is NP-complete in trading networks.

cs.CC↗

arXiv2018

Trading Networks with Bilateral Contracts

We consider a model of matching in trading networks in which firms can enter into bilateral contracts. In trading networks, stable outcomes, which are immune to deviations of arbitrary sets of firms, may not exist. We define a new solution concept called trail stability. Trail-stable outcomes are immune to consecutive, pairwise deviations between linked firms. We show that any trading network with bilateral contracts has a trail-stable outcome whenever firms' choice functions satisfy the full substitutability condition. For trail-stable outcomes, we prove results on the lattice structure, the rural hospitals theorem, strategy-proofness, and comparative statics of firm entry and exit. We also introduce weak trail stability which is implied by trail stability under full substitutability. We describe relationships between the solution concepts.

cs.GT↗