We study information-theoretic phase transitions for the detectability of latent geometry in bipartite random geometric graphs (RGGs) with Gaussian $d$-dimensional latent vectors, while only a subset of edges carries latent information, determined by a random mask with i.i.d. $\mathsf{Bern}(q)$ entries. For any fixed edge density $p \in (0,1)$, we determine essentially tight thresholds for this problem as a function of $d$ and $q$. Our results show that the detection problem is substantially easier if the mask is known up-front, compared to the case where the mask is hidden. Our analysis is built upon a novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs that exploits cancellations. The resulting bounds are applicable to much larger sub-graphs, which enables tight information-theoretic bounds for all noise levels instead, while the bounds considered in previous works only lead to lower bounds from the lens of low-degree polynomials. As a consequence, we identify the optimal information-theoretic thresholds and rule out computational–statistical gaps. Our bounds further improve upon the bounds on Fourier coefficients of random geometric graphs recently given by Bangachev and Bresler [STOC’24] in the dense bipartite case. We believe the techniques to extend to sparser and non-bipartite settings as well, at least if the considered sub-graphs are sufficiently small, and that they might help resolve open questions in related sparse or noisy detection problems.

We study efficient algorithms for recovering cliques in dense random intersection graphs (RIGs). In this model, $d = n^{\Omega(1)}$ cliques of size approximately $k$ are randomly planted by choosing the vertices to participate in each clique independently with probability $\delta$. While there has been extensive work on recovering one, or multiple disjointly planted cliques in random graphs, the natural extension of this question to recovering overlapping cliques has been, surprisingly, largely unexplored. Moreover, because every vertex can be part of polynomially many cliques, this task is significantly harder than in case of disjointly planted cliques (as recently studied by Kothari, Vempala, Wein and Xu [COLT’23]). In this work we obtain the first efficient algorithms for recovering the community structure of RIGs both from the perspective of exact and approximate recovery. Our algorithms are further robust to noise, monotone adversaries, a certain, optimal number of edge corruptions, and work whenever $k \gg \sqrt{n \log(n)}$. Our techniques follow the proofs-to-algorithms framework utilizing the sum-of-squares hierarchy. An essential component are certificates for the absence of large cliques outside of the ground-truth. Instead of spectral certificates, a central ingredient are modified versions of the biclique certificates, recently used for semi-random planted clique by Buhai, Kothari and Steurer [STOC’23]. To turn these certificates into robust and efficient algorithms that do not produce “false positives”, we rely on an extremely sharp concentration property for pseudo-distributions which might be of independent interest. Our techniques further extend to the related task of efficient \emph{refutation}, and lead to algorithms that can not only recover the ground-truth, but also certify the optimality of this clustering.

We aim to revive Thorup’s conjecture [Thorup, WG’92] on the existence of reachability shortcuts with ideal size-diameter tradeoffs. Thorup originally asked whether, given any graph G=(V,E) with m edges, we can add m1+o(1) “shortcut” edges E+ from the transitive closure E* of G so that G+(u,v) ≤ mo(1) for all (u,v)∈ E*, where G+=(V,E∪ E+). The conjecture was refuted by Hesse [Hesse, SODA’03], followed by significant efforts in the last few years to optimize the lower bounds.

Abstract We compare the $$(1, \lambda )$$ -EA and the $$(1 + \lambda )$$ -EA on the recently introduced benchmark $$\textsc {DisOM}$$ , which is the OneMax function with randomly planted local optima. Previous work showed that if all local optima have the same relative height, then the plus strategy never loses more than a factor $$O(n\log n)$$ compared to the comma strategy. Here we show that even small random fluctuations in the heights of the local optima have a devastating effect for the plus strategy and lead to superpolynomial time to achieve a prescribed fitness target. On the other hand, due to their ability to escape local optima, comma strategies are unaffected by the height of the local optima and remain efficient. Our results hold for a broad class of possible distortions and show that the plus strategy, but not the comma strategy, is generally deceived by sparse unstructured fluctuations of a smooth landscape.

We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between \say{sufficiently close} vertices with respect to an $L_q$-norm. In this setting, we focus on distinguishing an RGG from an Erdős–Rényi graph if both models have the same marginal edge probability $p$. So far, most results in the literature considered either spherical RGGs with $L_2$-distance or toroidal RGGs under $L_\infty$-distance. However, for general $L_q$-distances, many questions remain open, especially if $p$ is allowed to depend on $n$. The main reason for this is that RGGs under $L_q$-distances can not easily be represented as the logical \say{AND} of their 1-dimensional counterparts, as is the case for $L_\infty$ geometries. To overcome this difficulty, we devise a novel technique for quantifying the dependence between edges based on a modified version of Edgeworth expansions. Our technique yields the first tight algorithmic upper bounds for distinguishing toroidal RGGs under general $L_q$ norms from Erdős–Rényi graphs for any fixed $p$ and $q$. We achieve this by showing that the signed triangle statistic can distinguish the two models when $d\ll n^3p^3$ for the whole regime of edge probabilities $\frac{c}{n} Cite this Paper BibTeX @InProceedings{pmlr-v291-baguley25a, title = {Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions}, author = {Baguley, Samuel and G{\"o}bel, Andreas and Pappik, Marcus and Schiller, Leon}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {200--201}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/baguley25a/baguley25a.pdf}, url = {https://proceedings.mlr.press/v291/baguley25a.html}, abstract = {We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between \say{sufficiently close} vertices with respect to an $L_q$-norm. In this setting, we focus on distinguishing an RGG from an Erdős–Rényi graph if both models have the same marginal edge probability $p$. So far, most results in the literature considered either spherical RGGs with $L_2$-distance or toroidal RGGs under $L_\infty$-distance. However, for general $L_q$-distances, many questions remain open, especially if $p$ is allowed to depend on $n$. The main reason for this is that RGGs under $L_q$-distances can not easily be represented as the logical \say{AND} of their 1-dimensional counterparts, as is the case for $L_\infty$ geometries. To overcome this difficulty, we devise a novel technique for quantifying the dependence between edges based on a modified version of Edgeworth expansions. Our technique yields the first tight algorithmic upper bounds for distinguishing toroidal RGGs under general $L_q$ norms from Erdős–Rényi graphs for any fixed $p$ and $q$. We achieve this by showing that the signed triangle statistic can distinguish the two models when $d\ll n^3p^3$ for the whole regime of edge probabilities $\frac{c}{n} Copy to Clipboard Download Endnote %0 Conference Paper %T Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions %A Samuel Baguley %A Andreas Göbel %A Marcus Pappik %A Leon Schiller %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-baguley25a %I PMLR %P 200--201 %U https://proceedings.mlr.press/v291/baguley25a.html %V 291 %X We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between \say{sufficiently close} vertices with respect to an $L_q$-norm. In this setting, we focus on distinguishing an RGG from an Erdős–Rényi graph if both models have the same marginal edge probability $p$. So far, most results in the literature considered either spherical RGGs with $L_2$-distance or toroidal RGGs under $L_\infty$-distance. However, for general $L_q$-distances, many questions remain open, especially if $p$ is allowed to depend on $n$. The main reason for this is that RGGs under $L_q$-distances can not easily be represented as the logical \say{AND} of their 1-dimensional counterparts, as is the case for $L_\infty$ geometries. To overcome this difficulty, we devise a novel technique for quantifying the dependence between edges based on a modified version of Edgeworth expansions. Our technique yields the first tight algorithmic upper bounds for distinguishing toroidal RGGs under general $L_q$ norms from Erdős–Rényi graphs for any fixed $p$ and $q$. We achieve this by showing that the signed triangle statistic can distinguish the two models when $d\ll n^3p^3$ for the whole regime of edge probabilities $\frac{c}{n} Copy to Clipboard Download APA Baguley, S., Göbel, A., Pappik, M. & Schiller, L.. (2025). Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:200-201 Available from https://proceedings.mlr.press/v291/baguley25a.html. Copy to Clipboard Download Related Material Download PDF This site last compiled Sun, 05 Jul 2026 15:03:27 +0000 Github Account Copyright © The authors and PMLR 2026. MLResearchPress

In this paper, we study graph distances in the geometric random graph models scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and hyperbolic random graphs HRG. Despite the wide success of the models, the parameter regime in which graph distances are polylogarithmic is poorly understood. We provide new and improved lower bounds. In a certain portion of the parameter regime, those match the known upper bounds. Compared to the best previous lower bounds by Hao and Heydenreich, our result has several advantages: it gives matching bounds for a larger range of parameters, thus settling the question for a larger portion of the parameter space. It strictly improves the lower bounds by Hao and Heydenreich for all parameters settings in which those bounds were not tight. It gives tail bounds on the probability of having short paths, which imply shape theorems for the $k$-neighbourhood of a vertex whenever our lower bounds are tight, and tight bounds for the size of this $k$-neighbourhood. And last but not least, our proof is much simpler and not much longer than two pages, and we demonstrate that it generalizes well by showing that the same technique also works for first passage percolation.

We compare the (1, λ)-EA and the (1 + λ)-EA on the recently introduced benchmark DisOM, which is the OneMax function with randomly planted local optima. Previous work showed that if all local optima have the same relative height, then the plus strategy never loses more than a factor O(n log n) compared to the comma strategy. Here we show that even small random fluctuations in the heights of the local optima have a devastating effect for the plus strategy and lead to super-polynomial runtimes. On the other hand, due to their ability to escape local optima, comma strategies are unaffected by the height of the local optima and remain efficient. Our results hold for a broad class of possible distortions and show that the plus strategy, but not the comma strategy, is generally deceived by sparse unstructured fluctuations of a smooth landscape. We further develop new techniques for analyzing "frozen noise" which may be of independent interest.

Abstract. A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach nongeometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the multiple-routes (MRs) problem, which is given a start and destination and aims at finding up to$n$different routes that the drivers strategically disperse over, minimizing the overall travel time of the system. Due to theNP-hard nature of the problem, we introduce the MRs evolutionary algorithm (MREA) as a heuristic solver. We study several mutation and crossover operators and evaluate them on real-world data of Berlin, Germany. We find that a combination of all operators yields the best result, reducing the overall travel time by a factor between 1.8 and 3, in the median, compared to all drivers taking the fastest route. 6mm]Please cite reference[2]in the text of the paper. It was removed from the abstract as having reference in an abstract is contrary to IEEE journal style.For the base case$n=2$, we compare our MREA to the highly tailored optimal solver by Bläsius et al. (2020), and show that, in the median, our approach finds solutions of quality at least 99.69% of an optimal solution while only requiring 40% of the time.

To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1-bits allowed in the length- n bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where B bits have a weight of 1+ 1/ n and the remaining bits have a weight of 1. Friedrich et al. [TCS 2020] gave a bound of Θ ( n 2 ) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a 1 and a 0 simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n ) if n-B = O (1). However, if n-B = Θ ( n ), we show a bound of Ω ( n 2 ), just as for classic bit mutation. Crossover together with a simple island model gives running times of O ( n 2 / log n ) (uniform crossover) and \(O(n\sqrt {n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity, we show a bound of O ( n log n ). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.

In order to understand better how and why crossover can benefit optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1s allowed in the bit string (cardinality constraint). We consider the natural translation of the OneMax test function, a linear function where B bits have a weight of 1 + ε and the remaining bits have a weight of 1. The literature gives a bound of Θ(n2) for the (1+1) EA on this function.

Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the Multiple-Routes problem, which is given a start and destination and aims at finding up to n different routes that the drivers strategically disperse over, minimizing the overall travel time of the system.

Traditional navigation services find the fastest route for a single driver. Though always using the fastest route seems desirable for every individual, selfish behavior can have undesirable effects such as higher energy consumption and avoidable congestion, even leading to higher overall and individual travel times. In contrast, strategic routing aims at optimizing the traffic for all agents regarding a global optimization goal. We introduce a framework to formalize real-world strategic routing scenarios as algorithmic problems and study one of them, which we call Single Alternative Path (SAP), in detail. There, we are given an original route between a single origin--destination pair. The goal is to suggest an alternative route to all agents that optimizes the overall travel time under the assumption that the agents distribute among both routes according to a psychological model, for which we introduce the concept of Pareto-conformity. We show that the SAP problem is NP-complete, even for such models. Nonetheless, assuming Pareto-conformity, we give multiple algorithms for different variants of SAP, using multi-criteria shortest path algorithms as subroutines. Moreover, we prove that several natural models are in fact Pareto-conform. The implementation of our algorithms serves as a proof of concept, showing that SAP can be solved in reasonable time even though the algorithms have exponential running time in the worst case.