arXiv subjects

Francesco d'Amore

Publications and source records attributed to Francesco d'Amore.

At least 19 recordsLinked to original sources

arXiv2026

Distributed Quantum Advantage in Locally Checkable Labeling Problems

In this paper, we present the first known example of a locally checkable labeling problem (LCL) that admits asymptotic distributed quantum advantage in the LOCAL model of distributed computing: our problem can be solved in $O(\log n)$ communication rounds in the quantum-LOCAL model, but it requires $Ω(\log n \cdot \log^{0.99} \log n)$ communication rounds in the classical randomized-LOCAL model. We also show that distributed quantum advantage cannot be arbitrarily large: if an LCL problem can be solved in $T(n)$ rounds in the quantum-LOCAL model, it can also be solved in $\tilde O(\sqrt{n T(n)})$ rounds in the classical randomized-LOCAL model. In particular, a problem that is strictly global classically is also almost-global in quantum-LOCAL. Our second result also holds for $T(n)$-dependent probability distributions. As a corollary, if there exists a finitely dependent distribution over valid labelings of some LCL problem $Π$, then the same problem $Π$ can also be solved in $\tilde O(\sqrt{n})$ rounds in the classical randomized-LOCAL and deterministic-LOCAL models. That is, finitely dependent distributions cannot exist for global LCL problems.

cs.DC↗

arXiv2026

Superlogarithmic Gap Result for LCLs on Trees in Quantum-LOCAL

We show that, on trees, any locally checkable labeling problem (LCL) $Π$ that can be solved by an $n^{o(1)}$-dependent distribution can also be solved by an $O(\log n)$-round deterministic LOCAL algorithm. The result is obtained through a rake-and-compress-style decomposition of the input tree, and local simulations of the bounded dependent distribution on the components of the decomposition. As a corollary to our result, any LCL problem on trees can either be solved by an $O(\log n)$ deterministic LOCAL algorithm, or requires $n^{Ω(1)}$ rounds to solve by a quantum-LOCAL algorithm.

cs.CC↗

arXiv2026

New Hardness Results for the LOCAL Model via a Simple Self-Reduction

Very recently, Khoury and Schild [FOCS 2025] showed that any randomized LOCAL algorithm that solves maximal matching requires $Ω(\min\{\log Δ, \log_Δn\})$ rounds, where $n$ is the number of nodes in the graph and $Δ$ is the maximum degree. This result is shown through a new technique, called round elimination via self-reduction. The lower bound proof is beautiful and presents very nice ideas. However, it spans more than 25 pages of technical details, and hence it is hard to digest and generalize to other problems. Historically, the simplification of proofs and techniques has marked an important turning point in our understanding of the complexity of graph problems. Our paper makes a step forward towards this direction, and provides the following contributions. 1. We present a short and simplified version of the round elimination via self-reduction technique. The simplification of this technique enables us to obtain the following two hardness results. 2. We show that any randomized LOCAL algorithm that solves the maximal $b$-matching problem requires $Ω(\min\{\log_{1+b}Δ, \log_Δn\})$ and $Ω(\sqrt{\log_{1+b} n})$ rounds. We recall that the $b$-matching problem is a generalization of the matching problem where each vertex can have up to $b$ incident edges in the matching. As a corollary, for $b=1$, we obtain a short proof for the maximal matching lower bound shown by Khoury and Schild. 3. We show that any randomized LOCAL algorithm that properly colors the edges of a graph with $Δ+ k$ colors requires $Ω(\min\{\log Δ, \log_Δn\})$ and $Ω(\sqrt{\log n})$ rounds, for any $k\le Δ^{1-\varepsilon}$ and any constant $\varepsilon > 0$.

cs.DC↗

arXiv2026

DejaVu: A Minimalistic Mechanism for Distributed Plurality Consensus

We study the plurality consensus problem in distributed systems where a population of extremely simple agents, each initially holding one of $k$ opinions, aims to agree on the initially most frequent one. In this setting, $h$-majority is arguably the simplest and most studied protocol, in which each agent samples the opinion of $h$ neighbors uniformly at random and updates its opinion to the most frequent value in the sample. We propose a new, extremely simple mechanism called DéjàVu: an agent queries neighbors until it encounters an opinion for the second time, at which point it updates its own opinion to the duplicate value. This rule does not require agents to maintain counters or estimate frequencies, nor to choose any parameter (such as a sample size $h$); it relies solely on the primitive ability to detect repetition. We provide a rigorous analysis of DéjàVu that relies on several technical ideas of independent interest and demonstrates that it is competitive with $h$-majority and, in some regimes, substantially more communication-efficient, thus yielding a powerful primitive for plurality consensus.

cs.DC↗

arXiv2026

Polynomially Over-Parameterized Convolutional Neural Networks Contain Structured Strong Winning Lottery Tickets

The Strong Lottery Ticket Hypothesis (SLTH) states that randomly-initialised neural networks likely contain subnetworks that perform well without any training. Although unstructured pruning has been extensively studied in this context, its structured counterpart, which can deliver significant computational and memory efficiency gains, has been largely unexplored. One of the main reasons for this gap is the limitations of the underlying mathematical tools used in formal analyses of the SLTH. In this paper, we overcome these limitations: we leverage recent advances in the multidimensional generalisation of the Random Subset-Sum Problem and obtain a variant that admits the stochastic dependencies that arise when addressing structured pruning in the SLTH. We apply this result to prove, for a wide class of random Convolutional Neural Networks, the existence of structured subnetworks that can approximate any sufficiently smaller network. This result provides the first sub-exponential bound around the SLTH for structured pruning, opening up new avenues for further research on the hypothesis and contributing to the understanding of the role of over-parameterization in deep learning.

cs.LG↗

arXiv2026

Distributed Algorithms for Potential Problems

In this work, we present a fast distributed algorithm for local potential problems: these are graph problems where the task is to find a locally optimal solution where no node can unilaterally improve the utility in its local neighborhood by changing its own label. A simple example of such a problem is the task of finding a locally optimal cut, i.e., a cut where for each node at least half of its incident edges are cut edges. The distributed round complexity of the locally optimal cut problem has been wide open; the problem is known to require $Ω(\log n)$ rounds in the deterministic LOCAL model and $Ω(\log \log n)$ rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of $O(n)$ rounds. Locally optimal cut in constant-degree graphs is perhaps the simplest example of a locally checkable labeling problem for which there is still such a large gap between current upper and lower bounds. We show that in constant-degree graphs, all local potential problems, including locally optimal cut, can be solved in $\log^{O(1)} n$ rounds, both in the deterministic and randomized LOCAL models. In particular, the deterministic round complexity of the locally optimal cut problem is now settled to $\log^{Θ(1)} n$. Our algorithms also apply to the general case of graphs of maximum degree $Δ$. For the special case of locally optimal cut, we obtain a randomized algorithm that runs in $O(Δ^{2} \log^{6} n)$ rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in $Δ$ is necessary: we prove a lower bound of $Ω(\min\{Δ,\sqrt{n}\})$ rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.

cs.DC↗

arXiv2026

Online Locality Meets Distributed Quantum Computing

We connect three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing: A. distributed quantum computing and non-signaling distributions [e.g. STOC 2024], B. finitely-dependent processes [e.g. Forum Math. Pi 2016], and C. locality in online graph algorithms and dynamic graph algorithms [e.g. ICALP 2023]. We prove new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs). We show that all these settings can be sandwiched between the classical LOCAL model and what we call the randomized online-LOCAL model. Our work implies limitations on the quantum advantage in the distributed setting, and we also exhibit a new barrier for proving tighter bounds. Our main technical results are these: 1. All LCL problems solvable with locality $O(\log^\star n)$ in the classical deterministic LOCAL model admit a finitely-dependent distribution with locality $O(1)$. This answers an open question by Holroyd [2024], and also presents a new barrier for proving bounds on distributed quantum advantage using causality-based arguments. 2. In rooted trees, if we can solve an LCL problem with locality $o(\log \log \log n)$ in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. One of many implications is that in rooted trees, $O(\log^\star n)$ locality in quantum-LOCAL is not stronger than $O(\log^\star n)$ locality in classical LOCAL.

cs.DC↗

arXiv2025

New Limits on Distributed Quantum Advantage: Dequantizing Linear Programs

In this work, we give two results that put new limits on distributed quantum advantage in the context of the LOCAL model of distributed computing. First, we show that there is no distributed quantum advantage for any linear program. Put otherwise, if there is a quantum-LOCAL algorithm $\mathcal{A}$ that finds an $α$-approximation of some linear optimization problem $Π$ in $T$ communication rounds, we can construct a classical, deterministic LOCAL algorithm $\mathcal{A}'$ that finds an $α$-approximation of $Π$ in $T$ rounds. As a corollary, all classical lower bounds for linear programs, including the KMW bound, hold verbatim in quantum-LOCAL. Second, using the above result, we show that there exists a locally checkable labeling problem (LCL) for which quantum-LOCAL is strictly weaker than the classical deterministic SLOCAL model. Our results extend from quantum-LOCAL also to finitely dependent and non-signaling distributions, and one of the corollaries of our work is that the non-signaling model and the SLOCAL model are incomparable in the context of LCL problems: By prior work, there exists an LCL problem for which SLOCAL is strictly weaker than the non-signaling model, and our work provides a separation in the opposite direction.

cs.DC↗

arXiv2025

On the $h$-majority dynamics with many opinions

We present the first upper bound on the convergence time to consensus of the well-known $h$-majority dynamics with $k$ opinions, in the synchronous setting, for $h$ and $k$ that are both non-constant values. We suppose that, at the beginning of the process, there is some initial additive bias towards some plurality opinion, that is, there is an opinion that is supported by $x$ nodes while any other opinion is supported by strictly fewer nodes. We prove that, with high probability, if the bias is $ω(\sqrt{x})$ and the initial plurality opinion is supported by at least $x = ω(\log n)$ nodes, then the process converges to plurality consensus in $O(\log n)$ rounds whenever $h = ω(n \log n / x)$. A main corollary is the following: if $k = o(n / \log n)$ and the process starts from an almost-balanced configuration with an initial bias of magnitude $ω(\sqrt{n/k})$ towards the initial plurality opinion, then any function $h = ω(k \log n)$ suffices to guarantee convergence to consensus in $O(\log n)$ rounds, with high probability. Our upper bound shows that the lower bound of $Ω(k / h^2)$ rounds to reach consensus given by Becchetti et al. (2017) cannot be pushed further than $\widetildeΩ(k / h)$. Moreover, the bias we require is asymptotically smaller than the $Ω(\sqrt{n\log n})$ bias that guarantees plurality consensus in the $3$-majority dynamics: in our case, the required bias is at most any (arbitrarily small) function in $ω(\sqrt{x})$ for any value of $k \ge 2$.

cs.DC↗

arXiv2025

On the Limits of Distributed Quantum Computing

Quantum advantage is well-established in centralized computing, where quantum algorithms can solve certain problems exponentially faster than classical ones. In the distributed setting, significant progress has been made in bandwidth-limited networks, where quantum distributed networks have shown computational advantages over classical counterparts. However, the potential of quantum computing in networks that are constrained only by large distances is not yet understood. We focus on the LOCAL model of computation (Linial, FOCS 1987), a distributed computational model where computational power and communication bandwidth are unconstrained, and its quantum generalization. In this brief survey, we summarize recent progress on the quantum-LOCAL model outlining its limitations with respect to its classical counterpart: we discuss emerging techniques, and highlight open research questions that could guide future efforts in the field.

cs.DC↗

arXiv2024

Phase transition of the 3-majority opinion dynamics with noisy interactions

Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. In particular, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics that are not strongly affected by noisy communications. In this work, we study the popular 3-Majority dynamics, an opinion dynamics which has been proved to be an efficient protocol for the majority consensus problem, in which we introduce a simple feature of uniform communication noise, following (d'Amore et al. 2020). We prove that in the fully connected communication network of n agents and in the binary opinion case, the process induced by the 3-Majority dynamics exhibits a phase transition. For a noise probability $p < 1/3$, the dynamics reaches in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. Furthermore, departing from previous analyses, we further characterize this phase by showing that there exists an attractive equilibrium value $s_{\text{eq}} \in [n]$ for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, the agreement opinion turns out to be the initial majority one if the bias towards it is of magnitude $Ω(\sqrt{n\log n})$ in the initial configuration. If, instead, $p > 1/3$, no form of consensus is possible, and any information regarding the initial majority opinion is lost in logarithmic time with high probability. Despite more communications per-round are allowed, the 3-Majority dynamics surprisingly turns out to be less resilient to noise than the Undecided-State dynamics (d'Amore et al. 2020), whose noise threshold value is $p = 1/2$.

cs.DC↗

arXiv2024

Distributed Quantum Advantage for Local Problems

We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree $Δ$, any classical (deterministic or randomized) LOCAL model algorithm will require $Ω(Δ)$ rounds to solve the iterated GHZ problem, while the problem can be solved in $1$ round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires $Ω(Δ)$ rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

cs.DC↗

arXiv2024

No distributed quantum advantage for approximate graph coloring

We give an almost complete characterization of the hardness of $c$-coloring $χ$-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that: 1) We give a new distributed algorithm that finds a $c$-coloring in $χ$-chromatic graphs in $\tilde{\mathcal{O}}(n^{\frac{1}α})$ rounds, with $α= \bigl\lfloor\frac{c-1}{χ- 1}\bigr\rfloor$. 2) We prove that any distributed algorithm for this problem requires $Ω(n^{\frac{1}α})$ rounds. Our upper bound holds in the classical, deterministic LOCAL model, while the near-matching lower bound holds in the non-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, captures all models of distributed graph algorithms that obey physical causality; this includes not only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL, even with a pre-shared quantum state. We also show that similar arguments can be used to prove that, e.g., 3-coloring 2-dimensional grids or $c$-coloring trees remain hard problems even for the non-signaling model, and in particular do not admit any quantum advantage. Our lower-bound arguments are purely graph-theoretic at heart; no background on quantum information theory is needed to establish the proofs.

cs.DC↗

arXiv2024

Search via Parallel Lévy Walks on $\mathbb{Z}^2$

Motivated by the Lévy foraging hypothesis -- the premise that various animal species have adapted to follow Lévy walks to optimize their search efficiency -- we study the parallel hitting time of Lévy walks on the infinite two-dimensional grid. We consider $k$ independent discrete-time Lévy walks, with the same exponent $α\in(1,\infty)$, that start from the same node, and analyze the number of steps until the first walk visits a given target at distance $\ell$. We show that for any choice of $k$ and $\ell$ from a large range, there is a unique optimal exponent $α_{k,\ell} \in (2,3)$, for which the hitting time is $\tilde O(\ell^2/k)$ w.h.p., while modifying the exponent by an $ε$ term increases the hitting time by a polynomial factor, or the walks fail to hit the target almost surely. Based on that, we propose a surprisingly simple and effective parallel search strategy, for the setting where $k$ and $\ell$ are unknown: the exponent of each Lévy walk is just chosen independently and uniformly at random from the interval $(2,3)$. This strategy achieves optimal search time (modulo polylogarithmic factors) among all possible algorithms (even centralized ones that know $k$). Our results should be contrasted with a line of previous work showing that the exponent $α= 2$ is optimal for various search problems. In our setting of $k$ parallel walks, we show that the optimal exponent depends on $k$ and $\ell$, and that randomizing the choice of the exponents works simultaneously for all $k$ and $\ell$.

cs.DC↗

arXiv2023

Distributed derandomization revisited

One of the cornerstones of the distributed complexity theory is the derandomization result by Chang, Kopelowitz, and Pettie [FOCS 2016]: any randomized LOCAL algorithm that solves a locally checkable labeling problem (LCL) can be derandomized with at most exponential overhead. The original proof assumes that the number of random bits is bounded by some function of the input size. We give a new, simple proof that does not make any such assumptions-it holds even if the randomized algorithm uses infinitely many bits. While at it, we also broaden the scope of the result so that it is directly applicable far beyond LCL problems.

cs.DC↗

arXiv2023

Revisiting the Random Subset Sum problem

The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$, and we seek a subset of the $X_i$s whose sum approximates $z$ up to error $\varepsilon$. In this setup, it has been shown that, under mild assumptions on the distribution of the random variables, a sample of size $\mathcal{O}(\log(1/\varepsilon))$ suffices to obtain, with high probability, approximations for all values in $[-1/2, 1/2]$. Recently, this result has been rediscovered outside the algorithms community, enabling meaningful progress in other fields. In this work we present an alternative proof for this theorem, with a more direct approach and resourcing to more elementary tools.

math.PR↗

arXiv2022

On the Multidimensional Random Subset Sum Problem

In the Random Subset Sum Problem, given $n$ i.i.d. random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. Despite its simple statement, this problem is of fundamental interest to both theoretical computer science and statistical mechanics. More recently, it gained renewed attention for its implications in the theory of Artificial Neural Networks. An obvious multidimensional generalisation of the problem is to consider $n$ i.i.d. $d$-dimensional random vectors, with the objective of approximating every point $\mathbf{z} \in [-1,1]^d$. In 1998, G. S. Lueker showed that, in the one-dimensional setting, $n=\mathcal{O}(\log \frac 1\varepsilon)$ samples guarantee the approximation property with high probability.In this work, we prove that, in $d$ dimensions, $n = \mathcal{O}(d^3\log \frac 1\varepsilon \cdot (\log \frac 1\varepsilon + \log d))$ samples suffice for the approximation property to hold with high probability. As an application highlighting the potential interest of this result, we prove that a recently proposed neural network model exhibits universality: with high probability, the model can approximate any neural network within a polynomial overhead in the number of parameters.

cs.NE↗

arXiv2021

Planning with Biological Neurons and Synapses

We revisit the planning problem in the blocks world, and we implement a known heuristic for this task. Importantly, our implementation is biologically plausible, in the sense that it is carried out exclusively through the spiking of neurons. Even though much has been accomplished in the blocks world over the past five decades, we believe that this is the first algorithm of its kind. The input is a sequence of symbols encoding an initial set of block stacks as well as a target set, and the output is a sequence of motion commands such as "put the top block in stack 1 on the table". The program is written in the Assembly Calculus, a recently proposed computational framework meant to model computation in the brain by bridging the gap between neural activity and cognitive function. Its elementary objects are assemblies of neurons (stable sets of neurons whose simultaneous firing signifies that the subject is thinking of an object, concept, word, etc.), its commands include project and merge, and its execution model is based on widely accepted tenets of neuroscience. A program in this framework essentially sets up a dynamical system of neurons and synapses that eventually, with high probability, accomplishes the task. The purpose of this work is to establish empirically that reasonably large programs in the Assembly Calculus can execute correctly and reliably; and that rather realistic -- if idealized -- higher cognitive functions, such as planning in the blocks world, can be implemented successfully by such programs.

cs.AI↗