Emilio Cruciani

Emilio Cruciani, Sebastian Forster, Gramoz Goranci et al.

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to its closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on point sets in arbitrary metric spaces. In this paper, we give a deterministic decremental $(2+ε)$-approximation algorithm and a randomized incremental $(4+ε)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+ε)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art fully dynamic $(1+ε)$-approximation single-source shortest paths algorithm in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+ε)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.

Emilio Cruciani, Emanuela L. Giacomelli, Jinyeop Lee

Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multi-body interactions. Herein, we investigate a discrete-time nonlinear averaging dynamics with three-body interactions: an underlying hypergraph, comprising triples as hyperedges, delineates the structure of these interactions, while the vertices update their states through a weighted, state-dependent average of neighboring pairs' states. This dynamics captures reinforcing group effects, such as peer pressure, and exhibits higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converges to a multiplicatively-shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.

Emilio Cruciani, Sebastian Forster, Antonis Skarlatos

Given a weighted undirected graph, a number of clusters $k$, and an exponent $z$, the goal in the $(k, z)$-clustering problem on graphs is to select $k$ vertices as centers that minimize the sum of the distances raised to the power $z$ of each vertex to its closest center. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact $(k, z)$-clustering solution in the induced shortest-path metric. While efficient dynamic $k$-center approximation algorithms on graphs exist [Cruciani et al. SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic $(k,z)$-clustering problem. As the main result of this paper, we develop a randomized incremental $(k, z)$-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of $\tilde O(k m^{1+o(1)}+ k^{1+\frac{1}λ} m)$, where $λ\geq 1$ is an arbitrary fixed constant. Our incremental algorithm consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size $\tilde{O}(k)$ with a total update time of $m^{1+o(1)}$ over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant, a property that may be of independent interest. In the second stage, we maintain a constant-factor approximate $(k,z)$-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static $(k,z)$-clustering algorithm.

Emilio Cruciani, Roberto Verdecchia

Artificial Intelligence is increasingly pervasive across domains, with ever more complex models delivering impressive predictive performance. This fast technological advancement however comes at a concerning environmental cost, with state-of-the-art models - particularly deep neural networks and large language models - requiring substantial computational resources and energy. In this work, we present the intuition of Green AI dynamic model selection, an approach based on dynamic model selection that aims at reducing the environmental footprint of AI by selecting the most sustainable model while minimizing potential accuracy loss. Specifically, our approach takes into account the inference task, the environmental sustainability of available models, and accuracy requirements to dynamically choose the most suitable model. Our approach presents two different methods, namely Green AI dynamic model cascading and Green AI dynamic model routing. We demonstrate the effectiveness of our approach via a proof of concept empirical example based on a real-world dataset. Our results show that Green AI dynamic model selection can achieve substantial energy savings (up to ~25%) while substantially retaining the accuracy of the most energy greedy solution (up to ~95%). As conclusion, our preliminary findings highlight the potential that hybrid, adaptive model selection strategies withhold to mitigate the energy demands of modern AI systems without significantly compromising accuracy requirements.