arXiv subjects

Tatiana Rocha Avila

Publications and source records attributed to Tatiana Rocha Avila.

3 recordsLinked to original sources

arXiv2026

Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting

Recent concurrent work by Dupré la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math '81], it remains widely open whether comparable non-additive bounds can be achieved. Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the $n$ items covers only $t$ elements across all functions, we prove a constructive discrepancy bound that is polynomial in $t$, the number of colors $k$, and $\log n$.

math.CO↗

arXiv2026

Temporal Conductance and Bounds on the Voter Model for Dynamic Networks

The voter model is a classical stochastic process that models how opinions might spread through a network: at each step, every node lazily adopts the opinion of a random neighbour; eventually all nodes share the same opinion (consensus). Stronger connectivity should yield faster consensus. Berenbrink, Giakkoupis, Kermarrec, and Mallmann-Trenn (ICALP 2016) make this precise via the network's conductance: if the network has $m$ edges, minimum degree $d_{\min}$, and conductance at least $ϕ$, then the voter model reaches consensus in expected $O(m/(d_{\min}ϕ))$ steps. Their results extend to dynamic networks with fixed vertex degrees by considering the network's conductance at each time step. We introduce temporal conductance $Φ$, a more general connectivity measure for dynamic networks. Unlike static conductance, which collapses to $0$ whenever some snapshot is disconnected, $Φ$ captures connectivity through edges that appear at different times. We generalise the results of Berenbrink et al. from static conductance to temporal conductance, showing that the expected consensus time of the standard voter model is at most $O(m/(d_{\min}Φ))$. Moreover, we prove that this bound is tight up to constant factors. We expect temporal conductance to be a useful primitive for analysing other dynamics on temporal networks, and potentially time-inhomogeneous Markov chains more generally.

cs.DC↗

arXiv2026

On the Complexity of the Minimum-($k,ρ$)-Shortcut Problem

We consider the Minimum-$(k,ρ)$-$\mathrm{Shortcut}$ problem ($\min(k,ρ)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $ρ$ closest vertices using paths of at most $k$ edges. This is a fundamental graph optimization problem used to accelerate parallel shortest path algorithms. It is well-known that the problem is trivially solvable for the cases $k=1$ and $k\geqρ$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq 3$ if $ρ=Θ(n^ε)$, the boundary where the problem transitions from polynomial-time solvable to NP-hard remained open. In this paper, we narrow this gap significantly. We present a simpler and more direct reduction from the Hitting Set problem which establishes that $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq2$ and $ρ\geq k+2$ in both directed and undirected graphs. Complementing this, we use the symmetry of the undirected case to show that $ρ=k+1$ is solvable in polynomial time, a regime where the directed version remains a candidate for NP-hardness. Therefore, we obtain an almost complete characterization of the complexity of $\min(k,ρ)\text{-}\mathrm{Shortcut}$, with the sole remaining open case being $ρ= k+1$ in the directed setting.

cs.CC↗