arXiv subjects

Luca Trevisan

Publications and source records attributed to Luca Trevisan.

At least 19 recordsLinked to original sources

arXiv2026

Repurposing acquisition devices into trigger-based timing synchronization of breakdown events during MITICA high voltage holding experiments

A critical requirement for MITICA -- a full-scale prototype of the heating Neutral Beam Injectors hosted at the Consorzio RFX Neutral Beam Test Facility for the ITER experiment -- is the capability to withstand a continuous voltage of 1MV across the vacuum gaps insulating the beam source from the grounded vessel. To validate such feature, a dedicated voltage-holding test campaign was conducted throughout 2024 and 2025 using a full-scale mock-up of the beam source. The tests also involved an accurate characterization of the associated breakdown events: vacuum dielectric failures which result in rapid potential drops and generate strong current discharges. This contribution will present a relative time reconstruction architecture based on cost-effective, embedded RedPitaya (Zynq-7000 FPGA) devices repurposed as timing hubs. These nodes function as configurable trigger multiplexers while simultaneously recording trigger signals as transients to facilitate the offline reconstruction of event sequences. The method allows self-calibration through measuring the static intrinsic delays of the optical fibers and internal logics, generating delay offsets to synchronize acquired waveforms across a sparse, connected-graph topology of both acquisition devices and hubs themselves.

physics.comp-ph↗

arXiv2026

Recovering Communities in Structured Random Graphs

The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the $d$-dimensional hypercube graph admits $d$ distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability $p\in (0, 1)$? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate $p=C\log d/d$ for a sufficiently large constant $C$ is $1/\text{poly}(d)$-close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all $d$ cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger's cut counting argument.

cs.DS↗

arXiv2023

The Minority Dynamics and the Power of Synchronicity

We study the minority-opinion dynamics over a fully-connected network of $n$ nodes with binary opinions. Upon activation, a node receives a sample of opinions from a limited number of neighbors chosen uniformly at random. Each activated node then adopts the opinion that is least common within the received sample. Unlike all other known consensus dynamics, we prove that this elementary protocol behaves in dramatically different ways, depending on whether activations occur sequentially or in parallel. Specifically, we show that its expected consensus time is exponential in $n$ under asynchronous models, such as asynchronous GOSSIP. On the other hand, despite its chaotic nature, we show that it converges within $O(\log^2 n)$ rounds with high probability under synchronous models, such as synchronous GOSSIP. Finally, our results shed light on the bit-dissemination problem, that was previously introduced to model the spread of information in biological scenarios. Specifically, our analysis implies that the minority-opinion dynamics is the first stateless solution to this problem, in the parallel passive-communication setting, achieving convergence within a polylogarithmic number of rounds. This, together with a known lower bound for sequential stateless dynamics, implies a parallel-vs-sequential gap for this problem that is nearly quadratic in the number $n$ of nodes. This is in contrast to all known results for problems in this area, which exhibit a linear gap between the parallel and the sequential setting.

cs.DC↗

arXiv2023

New SDP Roundings and Certifiable Approximation for Cubic Optimization

We give new rounding schemes for SDP relaxations for the problems of maximizing cubic polynomials over the unit sphere and the $n$-dimensional hypercube. In both cases, the resulting algorithms yield a $O(\sqrt{n/k})$ multiplicative approximation in $2^{O(k)} \text{poly}(n)$ time. In particular, we obtain a $O(\sqrt{n/\log n})$ approximation in polynomial time. For the unit sphere, this improves on the rounding algorithms of Bhattiprolu et. al. [BGG+17] that need quasi-polynomial time to obtain a similar approximation guarantee. Over the $n$-dimensional hypercube, our results match the guarantee of a search algorithm of Khot and Naor [KN08] that obtains a similar approximation ratio via techniques from convex geometry. Unlike their method, our algorithm obtains an upper bound on the integrality gap of SDP relaxations for the problem and as a result, also yields a certificate on the optimum value of the input instance. Our results naturally generalize to homogeneous polynomials of higher degree and imply improved algorithms for approximating satisfiable instances of Max-3SAT. Our main motivation is the stark lack of rounding techniques for SDP relaxations of higher degree polynomial optimization in sharp contrast to a rich theory of SDP roundings for the quadratic case. Our rounding algorithms introduce two new ideas: 1) a new polynomial reweighting based method to round sum-of-squares relaxations of higher degree polynomial maximization problems, and 2) a general technique to compress such relaxations down to substantially smaller SDPs by relying on an explicit construction of certain hitting sets. We hope that our work will inspire improved rounding algorithms for polynomial optimization and related problems.

cs.DS↗

arXiv2023

A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs

We define a novel notion of ``non-backtracking'' matrix associated to any symmetric matrix, and we prove a ``Ihara-Bass'' type formula for it. We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with $k$ variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a $p$ fraction of assignments, if the instance contains $n$ variables and $n^{k/2} / ε^2$ constraints, we can efficiently compute a certificate that the optimum satisfies at most a $p+O_k(ε)$ fraction of constraints. Previously, this was known for even $k$, but for odd $k$ one needed $n^{k/2} (\log n)^{O(1)} / ε^2$ random constraints to achieve the same conclusion. Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck's inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is $o(n^{\lceil k/2 \rceil})$ and the third one cannot work when the number of constraints is $o(n^{k/2} \sqrt{\log n})$. We further apply our techniques to obtain a new PTAS finding assignments for $k$-CSP instances with $n^{k/2} / ε^2$ constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.

cs.CC↗

arXiv2023

On the Role of Memory in Robust Opinion Dynamics

We investigate opinion dynamics in a fully-connected system, consisting of $n$ identical and anonymous agents, where one of the opinions (which is called correct) represents a piece of information to disseminate. In more detail, one source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal for non-source agents is to quickly agree on this correct opinion, and do that robustly, i.e., from any initial configuration. The system evolves in rounds. In each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of $\ell$ random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. As restricted as it is, this setting encompasses very popular opinion dynamics, such as the voter model and best-of-$k$ majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that no dynamics can achieve correct convergence in an expected number of steps that is sub-quadratic in $n$, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case $\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may indeed require the use of memory. This insight may reflect on natural information dissemination processes that rely on a few knowledgeable individuals.

cs.DC↗

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↗

arXiv2022

Bond Percolation in Small-World Graphs with Power-Law Distribution

\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Full-bond percolation is also equivalent to the \emph{Reed-Frost process}, a network version of \emph{SIR} epidemic spreading, in which the graph represents contacts among people and $p$ corresponds to the probability that a contact between an infected person and a susceptible one causes a transmission of the infection. We consider \emph{one-dimensional power-law small-world graphs} with parameter $α$ obtained as the union of a cycle with additional long-range random edges: each pair of nodes $(u,v)$ at distance $L$ on the cycle is connected by a long-range edge $(u,v)$, with probability proportional to $1/L^α$. Our analysis determines three phases for the percolation subgraph $G_p$ of the small-world graph, depending on the value of $α$. 1) If $α< 1$, there is a $p<1$ such that, with high probability, there are $Ω(n)$ nodes that are reachable in $G_p$ from one another in $O(\log n)$ hops; 2) If $1 < α< 2$, there is a $p<1$ such that, with high probability, there are $Ω(n)$ nodes that are reachable in $G_p$ from one another in $\log^{O(1)}(n)$ hops; 3) If $α> 2$, for every $p<1$, with high probability all connected components of $G_p$ have size $O(\log n)$. The setting of full-bond percolation in finite graphs studied in this paper, which is the one that corresponds to the network SIR model of epidemic spreading, had not been analyzed before.

math.PR↗

arXiv2022

Percolation and Epidemic Processes in One-Dimensional Small-World Networks

We obtain tight thresholds for bond percolation on one-dimensional small-world graphs, and apply such results to obtain tight thresholds for the \emph{Independent Cascade} process and the \emph{Reed-Frost} process in such graphs. These are the first fully rigorous results establishing a phase transition for bond percolation and SIR epidemic processes in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative epidemiological phenomena emerge from our analysis, including the epidemic spread through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.

math.PR↗

arXiv2022

Spectral Robustness for Correlation Clustering Reconstruction in Semi-Adversarial Models

Correlation Clustering is an important clustering problem with many applications. We study the reconstruction version of this problem in which one is seeking to reconstruct a latent clustering that has been corrupted by random noise and adversarial modifications. Concerning the latter, there is a standard "post-adversarial" model in the literature, in which adversarial modifications come after the noise. Here, we introduce and analyse a "pre-adversarial" model in which adversarial modifications come before the noise. Given an input coming from such a semi-adversarial generative model, the goal is to reconstruct almost perfectly and with high probability the latent clustering. We focus on the case where the hidden clusters have nearly equal size and show the following. In the pre-adversarial setting, spectral algorithms are optimal, in the sense that they reconstruct all the way to the information-theoretic threshold beyond which no reconstruction is possible. This is in contrast to the post-adversarial setting, in which their ability to restore the hidden clusters stops before the threshold, but the gap is optimally filled by SDP-based algorithms. These results highlight a heretofore unknown robustness of spectral algorithms, showing them less brittle than previously thought.

cs.DS↗

arXiv2021

Cut Sparsification of the Clique Beyond the Ramanujan Bound: A Separation of Cut Versus Spectral Sparsification

We prove that a random $d$-regular graph, with high probability, is a cut sparsifier of the clique with approximation error at most $\left(2\sqrt{\frac 2 π} + o_{n,d}(1)\right)/\sqrt d$, where $2\sqrt{\frac 2 π} = 1.595\ldots$ and $o_{n,d}(1)$ denotes an error term that depends on $n$ and $d$ and goes to zero if we first take the limit $n\rightarrow \infty$ and then the limit $d \rightarrow \infty$. This is established by analyzing linear-size cuts using techniques of Jagannath and Sen derived from ideas in statistical physics, and analyzing small cuts via martingale inequalities. We also prove new lower bounds on spectral sparsification of the clique. If $G$ is a spectral sparsifier of the clique and $G$ has average degree $d$, we prove that the approximation error is at least the "Ramanujan bound'' $(2-o_{n,d}(1))/\sqrt d$, which is met by $d$-regular Ramanujan graphs, provided that either the weighted adjacency matrix of $G$ is a (multiple of) a doubly stochastic matrix, or that $G$ satisfies a certain high "odd pseudo-girth" property. The first case can be seen as an "Alon-Boppana theorem for symmetric doubly stochastic matrices," showing that a symmetric doubly stochastic matrix with $dn$ non-zero entries has a non-trivial eigenvalue of magnitude at least $(2-o_{n,d}(1))/\sqrt d$; the second case generalizes a lower bound of Srivastava and Trevisan, which requires a large girth assumption. Together, these results imply a separation between spectral sparsification and cut sparsification. If $G$ is a random $\log n$-regular graph on $n$ vertices, we show that, with high probability, $G$ admits a (weighted subgraph) cut sparsifier of average degree $d$ and approximation error at most $(1.595\ldots + o_{n,d}(1))/\sqrt d$, while every (weighted subgraph) spectral sparsifier of $G$ having average degree $d$ has approximation error at least $(2-o_{n,d}(1))/\sqrt d$.

cs.DS↗

arXiv2021

Average-Case Complexity

We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question whether the existence of hard-on-average problems in NP can be based on the P$\neq$NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case complexity. We discuss some of these "hardness amplification" results.

cs.CC↗

arXiv2021

Finding a Bounded-Degree Expander Inside a Dense One

It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $Δ$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also an expander. As with other consequences of the MSS theorem, it is not clear how one would explicitly construct such a subgraph. We show that such a subgraph (although with quantitatively weaker expansion and near-regularity properties than those predicted by MSS) can be constructed with high probability in linear time, via a simple algorithm. Our algorithm allows a distributed implementation that runs in $\mathcal O(\log n)$ rounds and does $\mathcal O(n)$ total work with high probability. The analysis of the algorithm is complicated by the complex dependencies that arise between edges and between choices made in different rounds. We sidestep these difficulties by following the combinatorial approach of counting the number of possible random choices of the algorithm which lead to failure. We do so by a compression argument showing that such random choices can be encoded with a non-trivial compression. Our algorithm bears some similarity to the way agents construct a communication graph in a peer-to-peer network, and, in the bipartite case, to the way agents select servers in blockchain protocols.

cs.DC↗

arXiv2020

Expansion and Flooding in Dynamic Random Networks with Node Churn

We study expansion and information diffusion in dynamic networks, that is in networks in which nodes and edges are continuously created and destroyed. We consider information diffusion by {\em flooding}, the process by which, once a node is informed, it broadcasts its information to all its neighbors. We study models in which the network is {\em sparse}, meaning that it has $\mathcal{O}(n)$ edges, where $n$ is the number of nodes, and in which edges are created randomly, rather than according to a carefully designed distributed algorithm. In our models, when a node is "born", it connects to $d=\mathcal{O}(1)$ random other nodes. An edge remains alive as long as both its endpoints do. If no further edge creation takes place, we show that, although the network will have $Ω_d(n)$ isolated nodes, it is possible, with large constant probability, to inform a $1-exp(-Ω(d))$ fraction of nodes in $\mathcal{O}(\log n)$ time. Furthermore, the graph exhibits, at any given time, a "large-set expansion" property. We also consider models with {\em edge regeneration}, in which if an edge $(v,w)$ chosen by $v$ at birth goes down because of the death of $w$, the edge is replaced by a fresh random edge $(v,z)$. In models with edge regeneration, we prove that the network is, with high probability, a vertex expander at any given time, and flooding takes $\mathcal{O}(\log n)$ time. The above results hold both for a simple but artificial streaming model of node churn, in which at each time step one node is born and the oldest node dies, and in a more realistic continuous-time model in which the time between births is Poisson and the lifetime of each node follows an exponential distribution.

cs.DC↗

arXiv2020

Lower bounds for Max-Cut in $H$-free graphs via semidefinite programming

For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming (SDP), to prove lower bounds on $f(G)$. We use this approach to find large cuts in graphs with few triangles and in $K_r$-free graphs.

cs.DS↗

arXiv2020

Subexponential LPs Approximate Max-Cut

We show that for every $\varepsilon > 0$, the degree-$n^\varepsilon$ Sherali-Adams linear program (with $\exp(\tilde{O}(n^\varepsilon))$ variables and constraints) approximates the maximum cut problem within a factor of $(\frac{1}{2}+\varepsilon')$, for some $\varepsilon'(\varepsilon) > 0$. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut, and hence resolves the extension complexity of approximate Max-Cut for approximation factors close to $\frac{1}{2}$ (up to the function $\varepsilon'(\varepsilon)$). Previously, only semidefinite programs and spectral methods were known to yield approximation factors better than $\frac 12$ for Max-Cut in time $2^{o(n)}$. We also show that constant-degree Sherali-Adams linear programs (with $\text{poly}(n)$ variables and constraints) can solve Max-Cut with approximation factor close to $1$ on graphs of small threshold rank: this is the first connection of which we are aware between threshold rank and linear programming-based algorithms. Our results separate the power of Sherali-Adams versus Lovász-Schrijver hierarchies for approximating Max-Cut, since it is known that $(\frac{1}{2}+\varepsilon)$ approximation of Max Cut requires $Ω_\varepsilon (n)$ rounds in the Lovász-Schrijver hierarchy. We also provide a subexponential time approximation for Khot's Unique Games problem: we show that for every $\varepsilon > 0$ the degree-$(n^\varepsilon \log q)$ Sherali-Adams linear program distinguishes instances of Unique Games of value $\geq 1-\varepsilon'$ from instances of value $\leq \varepsilon'$, for some $\varepsilon'( \varepsilon) >0$, where $q$ is the alphabet size. Such guarantees are qualitatively similar to those of previous subexponential-time algorithms for Unique Games but our algorithm does not rely on semidefinite programming or subspace enumeration techniques.

cs.DS↗

arXiv2019

A New Algorithm for the Robust Semi-random Independent Set Problem

In this paper, we study a general semi-random version of the planted independent set problem in a model initially proposed by Feige and Kilian, which has a large proportion of adversarial edges. We give a new deterministic algorithm that finds a list of independent sets, one of which, with high probability, is the planted one, provided that the planted set has size $k=Ω(n^{2/3})$. This improves on Feige and Kilian's original randomized algorithm, which with high probability recovers an independent set of size at least $k$ when $k=αn$ where $α$ is a constant.

cs.DS↗

arXiv2019

Vector Colorings of Random, Ramanujan, and Large-Girth Irregular Graphs

We prove that in sparse Erdős-Rényi graphs of average degree $d$, the vector chromatic number (the relaxation of chromatic number coming from the Lovàsz theta function) is typically $\tfrac{1}{2}\sqrt{d} + o_d(1)$. This fits with a long-standing conjecture that various refutation and hypothesis-testing problems concerning $k$-colorings of sparse Erdős-Rényi graphs become computationally intractable below the `Kesten-Stigum threshold' $d_{KS,k} = (k-1)^2$. Along the way, we use the celebrated Ihara-Bass identity and a carefully constructed non-backtracking random walk to prove two deterministic results of independent interest: a lower bound on the vector chromatic number (and thus the chromatic number) using the spectrum of the non-backtracking walk matrix, and an upper bound dependent only on the girth and universal cover. Our upper bound may be equivalently viewed as a generalization of the Alon-Boppana theorem to irregular graphs

cs.CC↗