arXiv subjects

Julia Gaudio

Publications and source records attributed to Julia Gaudio.

At least 19 recordsLinked to original sources

arXiv2026

An Improved Upper Bound for the Euclidean TSP Constant Using Band Crossovers

Consider $n$ points generated uniformly at random in the unit square, and let $L_n$ be the length of their optimal traveling salesman tour. Beardwood, Halton, and Hammersley (1959) showed $L_n / \sqrt n \to \beta$ almost surely as $n\to \infty$ for some constant $\beta$. The exact value of $\beta$ is unknown but estimated to be approximately $0.71$ (Applegate, Bixby, Chv\'atal, Cook 2011). Beardwood et al. further showed that $0.625 \leq \beta \leq 0.92116.$ Currently, the best known bounds are $0.6277 \leq \beta \leq 0.90380$, due to Gaudio and Jaillet (2019) and Carlsson and Yu (2023), respectively. The upper bound was derived using a computer-aided approach that is amenable to lower bounds with improved computation speed. In this paper, we show via simulation and concentration analysis that future improvement of the $0.90380$ is limited to $\sim0.88$. Moreover, we provide an alternative tour-constructing heuristic that, via simulation, could potentially improve the upper bound to $\sim0.85$. Our approach builds on a prior \emph{band-traversal} strategy, initially proposed by Beardwood et al. (1959) and subsequently refined by Carlsson and Yu (2023): divide the unit square into bands of height $\Theta(1/\sqrt{n})$, construct paths within each band, and then connect the paths to create a TSP tour. Our approach allows paths to cross bands, and takes advantage of pairs of points in adjacent bands which are close to each other. A rigorous numerical analysis improves the upper bound to $0.90367$.

cs.CG↗

arXiv2026

Exact Recovery in the Geometric Hidden Community Model

Hidden community problems, such as community detection in the Stochastic Block Model (SBM), submatrix localization, and $\mathbb{Z}_2$ synchronization, have received considerable attention in the probability, statistics, and information-theory literature. Motivated by transitive behavior in social networks, which tend to exhibit high triangle density, recent works have considered hidden community models in spatially-embedded networks. In particular, Baccelli and Sankararaman proposed the Geometric SBM, a spatially-embedded analogue of the standard SBM with dramatically more triangles. In this paper, we consider the problem of exact recovery for the Geometric Hidden Community Model (GHCM) of Gaudio, Guan, Niu, and Wei, which generalizes the Geometric SBM to allow for arbitrary pairwise observation distributions. Under mild technical assumptions, we find the information-theoretic threshold for exact recovery in the ``distance-dependent'' GHCM, which allows the pairwise distributions to depend on distance as well as community labels, thus completing the picture of exact recovery in spatially-embedded hidden community models.

math.PR↗

arXiv2025

Exact Recovery in the Geometric SBM

Community detection is the problem of identifying dense communities in networks. Motivated by transitive behavior in social networks ("thy friend is my friend"), an emerging line of work considers spatially-embedded networks, which inherently produce graphs containing many triangles. In this paper, we consider the problem of exact label recovery in the Geometric Stochastic Block Model (GSBM), a model proposed by Baccelli and Sankararaman as the spatially-embedded analogue of the well-studied Stochastic Block Model. Under mild technical assumptions, we completely characterize the information-theoretic threshold for exact recovery, generalizing the earlier work of Gaudio, Niu, and Wei.

math.PR↗

arXiv2025

Finding Planted Cycles in a Random Graph

In this paper, we study the problem of finding a collection of planted cycles in an \ER random graph $G \sim \mathcal{G}(n, \lambda/n)$, in analogy to the famous Planted Clique Problem. When the cycles are planted on a uniformly random subset of $\delta n$ vertices, we show that almost-exact recovery (that is, recovering all but a vanishing fraction of planted-cycle edges as $n \to \infty$) is information-theoretically possible if $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$ and impossible if $\lambda > \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. Moreover, despite the worst-case computational hardness of finding long cycles, we design a polynomial-time algorithm that attains almost exact recovery when $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. This stands in stark contrast to the Planted Clique Problem, where a significant computational-statistical gap is widely conjectured.

math.ST↗

arXiv2025

Semidefinite Programming for the Asymmetric Stochastic Block Model

We consider semidefinite programming (SDP) for the binary stochastic block model with equal-sized communities. Prior work of Hajek, Wu, and Xu proposed an SDP (sym-SDP) for the symmetric case where the intra-community edge probabilities are equal, and showed that the SDP achieves the information-theoretic threshold for exact recovery under the symmetry assumption. A key open question is whether SDPs can be used to achieve exact recovery for non-symmetric block models. In order to inform the design of a new SDP for the non-symmetric setting, we investigate the failure of sym-SDP when it is applied to non-symmetric settings. We formally show that sym-SDP fails to return the correct labeling of the vertices in some information-theoretically feasible, asymmetric cases. In addition, we give an intuitive geometric interpretation of the failure of sym-SDP in asymmetric settings, which in turn suggests an SDP formulation to handle the asymmetric setting. Still, this new SDP cannot be readily analyzed by existing techniques, suggesting a fundamental limitation in the design of SDPs for community detection.

cs.IT↗

arXiv2025

Phase Transitions in Planted k-Factor Recovery

This paper studies the problem of inferring a $k$-factor, specifically a spanning $k$-regular graph, planted within an Erdos-Renyi random graph $G(n,\lambda/n)$. We show that as the average degree $\lambda$ surpasses the critical threshold of $1/k$, the inference problem undergoes a transition from almost exact recovery to partial recovery. Moreover, as $\lambda$ tends to infinity, the accuracy of recovery diminishes to zero. In addition, we characterize the recovery accuracy of a linear-time iterative pruning algorithm and show that it achieves almost exact recovery when $\lambda < 1/k$. A key component of our analysis is a two-step cycle construction: we first build trees through local neighborhood exploration and then connect them by sprinkling using reserved edges. Interestingly, for proving impossibility of almost exact recovery, we construct $\Theta(n)$ many small trees of size $\Theta(1)$, whereas for establishing the algorithmic lower bound, a single large tree of size $\Theta(\sqrt{n\log n})$ suffices.

math.PR↗

arXiv2025

Sharp exact recovery threshold for two-community Euclidean random graphs

This paper considers the problem of label recovery in random graphs and matrices. Motivated by transitive behavior in real-world networks (i.e., ``the friend of my friend is my friend''), a recent line of work considers spatially-embedded networks, which exhibit transitive behavior. In particular, the Geometric Hidden Community Model (GHCM), introduced by Gaudio, Guan, Niu, and Wei, models a network as a labeled Poisson point process where every pair of vertices is associated with a pairwise observation whose distribution depends on the labels and positions of the vertices. The GHCM is in turn a generalization of the Geometric SBM (proposed by Baccelli and Sankararaman). Gaudio et al. provided a threshold below which exact recovery is information-theoretically impossible. Above the threshold, they provided a linear-time algorithm that succeeds in exact recovery under a certain ``distinctness-of-distributions'' assumption, which they conjectured to be unnecessary. In this paper, we partially resolve the conjecture by showing that the threshold is indeed tight for the two-community GHCM. We provide a two-phase, linear-time algorithm that explores the spatial graph in a data-driven manner in Phase I to yield an almost exact labeling, which is refined to achieve exact recovery in Phase II. Our results extend achievability to geometric formulations of well-known inference problems, such as the planted dense subgraph problem and submatrix localization, in which the distinctness-of-distributions assumption does not hold.

cs.SI↗

arXiv2024

Spectral Recovery in the Labeled SBM

We consider the problem of exact community recovery in the Labeled Stochastic Block Model (LSBM) with $k$ communities, where each pair of vertices is associated with a label from the set $\{0,1, \dots, L\}$. A pair of vertices from communities $i,j$ is given label $\ell$ with probability $p_{ij}^{(\ell)}$, and the goal is to recover the community partition. We propose a simple spectral algorithm for exact community recovery, and show that it achieves the information-theoretic threshold in the logarithmic-degree regime, under the assumption that the eigenvalues of certain parameter matrices are distinct and nonzero. Our results generalize recent work of Dhara, Gaudio, Mossel, and Sandon (2023), who showed that a spectral algorithm achieves the information-theoretic threshold in the Censored SBM, which is equivalent to the LSBM with $L = 2$. Interestingly, their algorithm uses eigenvectors from two matrix representations of the graph, while our algorithm uses eigenvectors from $L$ matrices.

math.ST↗

arXiv2024

Exact Label Recovery in Euclidean Random Graphs

In this paper, we propose a family of label recovery problems on weighted Euclidean random graphs. The vertices of a graph are embedded in $\mathbb{R}^d$ according to a Poisson point process, and are assigned to a discrete community label. Our goal is to infer the vertex labels, given edge weights whose distributions depend on the vertex labels as well as their geometric positions. Our general model provides a geometric extension of popular graph and matrix problems, including submatrix localization and $\mathbb{Z}_2$-synchronization, and includes the Geometric Stochastic Block Model (proposed by Sankararaman and Baccelli) as a special case. We study the fundamental limits of exact recovery of the vertex labels. Under a mild distinctness of distributions assumption, we determine the information-theoretic threshold for exact label recovery, in terms of a Chernoff-Hellinger divergence criterion. Impossibility of recovery below the threshold is proven by a unified analysis using a Cram\'er lower bound. Achievability above the threshold is proven via an efficient two-phase algorithm, where the first phase computes an almost-exact labeling through a local propagation scheme, while the second phase refines the labels. The information-theoretic threshold is dictated by the performance of the so-called genie estimator, which decodes the label of a single vertex given all the other labels. This shows that our proposed models exhibit the local-to-global amplification phenomenon.

cs.SI↗

arXiv2024

Exact Community Recovery under Side Information: Optimality of Spectral Algorithms

We study the problem of exact community recovery in general, two-community block models, in the presence of node-attributed $side$ $information$. We allow for a very general side information channel for node attributes, and for pairwise (edge) observations, consider both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, Submatrix Localization, and $\mathbb{Z}_2$-Synchronization as special cases. A recent work of Dreveton et al. 2024 characterized the information-theoretic limit of a very general exact recovery problem with side information. In this paper, we show algorithmic achievability in the above important cases by designing a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the pairwise observation matrix. Using the powerful tool of entrywise eigenvector analysis of Abbe et al. 2020, we show that our spectral algorithm can mimic the so called $genie$-$aided$ $estimators$, where the $i^{\mathrm{th}}$ genie-aided estimator optimally computes the estimate of the $i^{\mathrm{th}}$ label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.

cs.SI↗

arXiv2023

Exact Community Recovery in the Geometric SBM

We study the problem of exact community recovery in the Geometric Stochastic Block Model (GSBM), where each vertex has an unknown community label as well as a known position, generated according to a Poisson point process in $\mathbb{R}^d$. Edges are formed independently conditioned on the community labels and positions, where vertices may only be connected by an edge if they are within a prescribed distance of each other. The GSBM thus favors the formation of dense local subgraphs, which commonly occur in real-world networks, a property that makes the GSBM qualitatively very different from the standard Stochastic Block Model (SBM). We propose a linear-time algorithm for exact community recovery, which succeeds down to the information-theoretic threshold, confirming a conjecture of Abbe, Baccelli, and Sankararaman. The algorithm involves two phases. The first phase exploits the density of local subgraphs to propagate estimated community labels among sufficiently occupied subregions, and produces an almost-exact vertex labeling. The second phase then refines the initial labels using a Poisson testing procedure. Thus, the GSBM enjoys local to global amplification just as the SBM, with the advantage of admitting an information-theoretically optimal, linear-time algorithm.

cs.SI↗

arXiv2022

Average-case and smoothed analysis of graph isomorphism

We propose a simple and efficient local algorithm for graph isomorphism which succeeds for a large class of sparse graphs. This algorithm produces a low-depth canonical labeling, which is a labeling of the vertices of the graph that identifies its isomorphism class using vertices' local neighborhoods. Prior work by Czajka and Pandurangan showed that the degree profile of a vertex (i.e., the sorted list of the degrees of its neighbors) gives a canonical labeling with high probability when $n p_n = \omega( \log^{4}(n) / \log \log n )$ (and $p_{n} \leq 1/2$); subsequently, Mossel and Ross showed that the same holds when $n p_n = \omega( \log^{2}(n) )$. We first show that their analysis essentially cannot be improved: we prove that when $n p_n = o( \log^{2}(n) / (\log \log n)^{3} )$, with high probability there exist distinct vertices with isomorphic $2$-neighborhoods. Our first main result is a positive counterpart to this, showing that $3$-neighborhoods give a canonical labeling when $n p_n \geq (1+\delta) \log n$ (and $p_n \leq 1/2$); this improves a recent result of Ding, Ma, Wu, and Xu, completing the picture above the connectivity threshold. Our second main result is a smoothed analysis of graph isomorphism, showing that for a large class of deterministic graphs, a small random perturbation ensures that $3$-neighborhoods give a canonical labeling with high probability. While the worst-case complexity of graph isomorphism is still unknown, this shows that graph isomorphism has polynomial smoothed complexity.

math.PR↗

arXiv2022

Joint Facility and Demand Location Problem

In typical applications of facility location problems, the location of demand is assumed to be an input to the problem. The demand may be fixed or dynamic, but ultimately outside the optimizers control. In contrast, there are settings, especially in humanitarian contexts, in which the optimizer decides where to locate a demand node. In this work, we introduce an optimization framework for joint facility and demand location. As examples of our general framework, we extend the well-known k-median and k-center problems into joint facility and demand location problems (JFDLP) and formulate them as integer programs. We propose a local search heuristic based on network flow. We apply our heuristic to a hurricane evacuation response case study. Our results demonstrate the challenging nature of these simultaneous optimization problems, especially when there are many potential locations. The local search heuristic is most promising when the the number of potential locations is large, while the number of facility and demand nodes to be located is small.

math.OC↗

arXiv2022

Spectral Algorithms Optimally Recover Planted Sub-structures

Spectral algorithms are an important building block in machine learning and graph algorithms. We are interested in studying when such algorithms can be applied directly to provide optimal solutions to inference tasks. Previous works by Abbe, Fan, Wang and Zhong (2020) and by Dhara, Gaudio, Mossel and Sandon (2022) showed the optimality for community detection in the Stochastic Block Model (SBM), as well as in a censored variant of the SBM. Here we show that this optimality is somewhat universal as it carries over to other planted substructures such as the planted dense subgraph problem and submatrix localization problem, as well as to a censored version of the planted dense subgraph problem.

cs.DS↗

arXiv2022

The Power of Two Matrices in Spectral Algorithms for Community Recovery

Spectral algorithms are some of the main tools in optimization and inference problems on graphs. Typically, the graph is encoded as a matrix and eigenvectors and eigenvalues of the matrix are then used to solve the given graph problem. Spectral algorithms have been successfully used for graph partitioning, hidden clique recovery and graph coloring. In this paper, we study the power of spectral algorithms using two matrices in a graph partitioning problem. We use two different matrices resulting from two different encodings of the same graph and then combine the spectral information coming from these two matrices. We analyze a two-matrix spectral algorithm for the problem of identifying latent community structure in large random graphs. In particular, we consider the problem of recovering community assignments exactly in the censored stochastic block model, where each edge status is revealed independently with some probability. We show that spectral algorithms based on two matrices are optimal and succeed in recovering communities up to the information theoretic threshold. Further, we show that for most choices of the parameters, any spectral algorithm based on one matrix is suboptimal. The latter observation is in contrast to our prior works (2022a, 2022b) which showed that for the symmetric Stochastic Block Model and the Planted Dense Subgraph problem, a spectral algorithm based on one matrix achieves the information theoretic threshold. We additionally provide more general geometric conditions for the (sub)-optimality of spectral algorithms.

math.ST↗

arXiv2022

Community Detection in the Hypergraph SBM: Exact Recovery Given the Similarity Matrix

Community detection is a fundamental problem in network science. In this paper, we consider community detection in hypergraphs drawn from the $hypergraph$ $stochastic$ $block$ $model$ (HSBM), with a focus on exact community recovery. We study the performance of polynomial-time algorithms which operate on the $similarity$ $matrix$ $W$, where $W_{ij}$ reports the number of hyperedges containing both $i$ and $j$. Under this information model, while the precise information-theoretic limit is unknown, Kim, Bandeira, and Goemans derived a sharp threshold up to which the natural min-bisection estimator on $W$ succeeds. As min-bisection is NP-hard in the worst case, they additionally proposed a semidefinite programming (SDP) relaxation and conjectured that it achieves the same recovery threshold as the min-bisection estimator. In this paper, we confirm this conjecture. We also design a simple and highly efficient spectral algorithm with nearly linear runtime and show that it achieves the min-bisection threshold. Moreover, the spectral algorithm also succeeds in denser regimes and is considerably more efficient than previous approaches, establishing it as the method of choice. Our analysis of the spectral algorithm crucially relies on strong $entrywise$ bounds on the eigenvectors of $W$. Our bounds are inspired by the work of Abbe, Fan, Wang, and Zhong, who developed entrywise bounds for eigenvectors of symmetric matrices with independent entries. Despite the complex dependency structure in similarity matrices, we prove similar entrywise guarantees.

cs.SI↗

arXiv2022

Exact Community Recovery in Correlated Stochastic Block Models

We consider the problem of learning latent community structure from multiple correlated networks. We study edge-correlated stochastic block models with two balanced communities, focusing on the regime where the average degree is logarithmic in the number of vertices. Our main result derives the precise information-theoretic threshold for exact community recovery using multiple correlated graphs. This threshold captures the interplay between the community recovery and graph matching tasks. In particular, we uncover and characterize a region of the parameter space where exact community recovery is possible using multiple correlated graphs, even though (1) this is information-theoretically impossible using a single graph and (2) exact graph matching is also information-theoretically impossible. In this regime, we develop a novel algorithm that carefully synthesizes algorithms from the community recovery and graph matching literatures.

math.ST↗

arXiv2022

Shotgun Assembly of Erdos-Renyi Random Graphs

Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of \ER random graphs $G(n, p_n)$, where $p_n = n^{-α}$ for $0 < α< 1$. We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-$1$ neighborhoods, $G$ is exactly reconstructable for $0 < α< \frac{1}{3}$, but not reconstructable for $\frac{1}{2} < α< 1$. Given the collection of distance-$2$ neighborhoods, $G$ is exactly reconstructable for $α\in \left(0, \frac{1}{2}\right) \cup \left(\frac{1}{2}, \frac{3}{5}\right)$, but not reconstructable for $\frac{3}{4} < α< 1$.

math.PR↗