arXiv subjects

Júlia Baligács

Publications and source records attributed to Júlia Baligács.

7 recordsLinked to original sources

arXiv2026

Online and Incremental Fractional Vertex Cover on Trees

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

cs.DS↗

arXiv2026

Randomization Helps in Online Graph Exploration: Breaking the Deterministic Lower Bound on Cycles

In online graph exploration, introduced by Kalyanasundaram and Pruhs (1994), an agent must visit all vertices of an initially unknown weighted graph and return to its starting position, while the graph is revealed only locally at visited vertices. Although the problem has attracted considerable attention, previous work has focused exclusively on deterministic algorithms. Randomized strategies are often substantially harder to analyze because of a fundamental challenge inherent to exploration. In this work, we give the first positive result showing that randomization can improve competitive guarantees in online graph exploration. To this end, we focus on cycles, a simple graph class which nevertheless captures a key difficulty of online exploration. Our main contribution is $\textsc{RandHeavyTest}$, a randomized algorithm for online exploration of cycles whose competitive ratio we prove to be at most 1.315. This establishes a strict separation from the deterministic setting, where the optimal competitive ratio is $\thickapprox 1.366$, and thus gives the first provable advantage of randomization in online graph exploration. A key step towards this result is a new, simplified optimal deterministic algorithm, $\textsc{HeavyTest}$, whose formulation naturally suggests the randomized variant. We complement our upper bounds with lower bounds of 1.115 for arbitrary randomized algorithms and 1.207 for the natural class of so-called forward-greedy algorithms, which includes $\textsc{RandHeavyTest}$.

cs.DS↗

arXiv2026

A coarse block-cut tree theorem

We prove a coarse analogue of the classic fact that every graph can be decomposed along its cut-vertices into $2$-connected components. Precisely, we prove that for every graph $G$ and a positive integer $d$, $G$ admits a tree decomposition whose adhesion sets have weak diameter at most $3d+2$ so that no two vertices $u,v$ lying in the same bag can be separated by a set of weak diameter at most $d$ whose distance from $u$ and $v$ is more than $d$. By the Coarse Menger's Theorem for two paths, this condition admits also a dual formulation, phrased in terms of the existence of two paths that are far from each other and connect the vicinity of $u$ with the vicinity of $v$.

math.CO↗

arXiv2026

Inevitability of Polarization in Geometric Opinion Exchange

Polarization and unexpected correlations between opinions on diverse topics (including in politics, culture and consumer choices) are an object of sustained attention. However, numerous theoretical models do not seem to convincingly explain these phenomena. This paper is motivated by a recent line of work, studying models where polarization can be explained in terms of biased assimilation and geometric interplay between opinions on various topics. The agent opinions are represented as unit vectors on a multidimensional sphere and updated according to geometric rules. In contrast to previous work, we focus on the classical opinion exchange setting, where the agents update their opinions in discrete time steps, with a pair of agents interacting randomly at every step. The opinions are updated according to an update rule belonging to a general class. Our findings are twofold. First, polarization appears to be ubiquitous in the class of models we study, requiring only relatively modest assumptions reflecting biased assimilation. Second, there is a qualitative difference between two-dimensional dynamics on the one hand, and three or more dimensions on the other. Accordingly, we prove almost sure polarization for a large class of update rules in two dimensions. Then, we prove polarization in three and more dimensions in more limited cases and try to shed light on central difficulties that are absent in two dimensions.

cs.SI↗

arXiv2025

Geometric opinion exchange polarizes in every dimension

A recent line of work studies models of opinion exchange where agent opinions about $d$ topics are tracked simultaneously. The opinions are represented as vectors on the unit $(d-1)$-sphere, and the update rule is based on the overall correlation between the relevant vectors. The update rule reflects the assumption of biased assimilation, i.e., a pair of opinions is brought closer together if their correlation is positive and further apart if the correlation is negative. This model seems to induce the polarization of opinions into two antipodal groups. This is in contrast to many other known models which tend to achieve consensus. The polarization property has been recently proved for $d=2$, but the general case of $d \ge 3$ remained open. In this work, we settle the general case, using a more detailed understanding of the model dynamics and tools from the theory of random processes.

cs.SI↗

arXiv2024

The metric Menger problem

We study a generalization of the well-known disjoint paths problem which we call the metric Menger problem, denoted MM(r,k), where one is given two subsets of a graph and must decide whether they can be connected by $k$ paths of pairwise distance at least $r$. We prove that this problem is NP-complete for every $r\geq 3$ and $k\geq 2$ by giving a reduction from 3SAT. This resolves a conjecture recently stated by Georgakopoulos and Papasoglu. On the other hand, we show that the problem is in XP when parameterised by treewidth and maximum degree by observing that it is `locally checkable'. In the case $r\leq 3$, we prove that it suffices to parameterise by treewidth. We also state some open questions relating to this work.

math.CO↗

arXiv2024

A $(5/3+ε)$-Approximation for Tricolored Non-crossing Euclidean TSP

In the Tricolored Euclidean Traveling Salesperson problem, we are given~$k=3$ sets of points in the plane and are looking for disjoint tours, each covering one of the sets. Arora (1998) famously gave a PTAS based on ``patching'' for the case $k=1$ and, recently, Dross et al.~(2023) generalized this result to~$k=2$. Our contribution is a $(5/3+ε)$-approximation algorithm for~$k=3$ that further generalizes Arora's approach. It is believed that patching is generally no longer possible for more than two tours. We circumvent this issue by either applying a conditional patching scheme for three tours or using an alternative approach based on a weighted solution for $k=2$.

cs.DS↗