This investigation is firstly focused into showing that two metric parameters represent the same object in graph theory. That is, we prove that the multiset resolving sets and the ID-colorings of graphs are the same thing. We also consider some computational and combinatorial problems of the multiset dimension, or equivalently, the ID-number of graphs. We prove that the decision problem concerning finding the multiset dimension of graphs is NP-complete. We consider the multiset dimension of king grids and prove that it is bounded above by 4. We also give a characterization of the strong product graphs with one factor being a complete graph, and whose multiset dimension is not infinite.
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A countable structure is indivisible if for every coloring with finite range there is a monochromatic isomorphic subcopy of the structure. Each indivisible structure $\mathcal{S}$ naturally corresponds to an indivisibility problem $\mathsf{Ind}\ \mathcal{S}$, which outputs such a subcopy given a presentation and coloring. We investigate the Weihrauch complexity of the indivisibility problems for two structures: the rational numbers $\mathbb{Q}$ as a linear order, and the equivalence relation $\mathscr{E}$ with countably many equivalence classes each having countably many members. We separate the Weihrauch degrees of both $\mathsf{Ind}\ \mathbb{Q}$ and $\mathsf{Ind}\ \mathscr{E}$ from several benchmark problems, showing in particular that $\mathsf{C}_\mathbb{N} \vert_\mathrm{W} \mathsf{Ind}\ \mathbb{Q}$ and hence $\mathsf{Ind}\ \mathbb{Q}$ is strictly weaker than the problem of finding an interval in which some color is dense for a given coloring of $\mathbb{Q}$; and that the Weihrauch degree of $\mathsf{Ind}\ \mathscr{E}_k$ is strictly between those of $\mathsf{SRT}^2_k$ and $\mathsf{RT}^2_k$, where $\mathsf{Ind}\ \mathcal{S}_k$ is the restriction of $\mathsf{Ind}\ \mathcal{S}$ to $k$-colorings.
Adversarial generative models, such as Generative Adversarial Networks (GANs), are widely applied for generating various types of data, i.e., images, text, and audio. Accordingly, its promising performance has led to the GAN-based adversarial attack methods in the white-box and black-box attack scenarios. The importance of transferable black-box attacks lies in their ability to be effective across different models and settings, more closely aligning with real-world applications. However, it remains challenging to retain the performance in terms of transferable adversarial examples for such methods. Meanwhile, we observe that some enhanced gradient-based transferable adversarial attack algorithms require prolonged time for adversarial sample generation. Thus, in this work, we propose a novel algorithm named GE-AdvGAN to enhance the transferability of adversarial samples whilst improving the algorithm's efficiency. The main approach is via optimising the training process of the generator parameters. With the functional and characteristic similarity analysis, we introduce a novel gradient editing (GE) mechanism and verify its feasibility in generating transferable samples on various models. Moreover, by exploring the frequency domain information to determine the gradient editing direction, GE-AdvGAN can generate highly transferable adversarial samples while minimizing the execution time in comparison to the state-of-the-art transferable adversarial attack algorithms. The performance of GE-AdvGAN is comprehensively evaluated by large-scale experiments on different datasets, which results demonstrate the superiority of our algorithm. The code for our algorithm is available at: //github.com/LMBTough/GE-advGAN
We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains obtained as the union of squared elements out of a uniform structured mesh, such as the one that naturally arises when the domain is issued from an image. We show, both theoretically and numerically, that resorting to the use of polygonal elements allows to satisfy, for any order, the assumptions required for the stability of the method, thus allowing to fully exploit the potential of higher order methods, the efficiency of which is ensured by a novel static condensation strategy acting on the edges of the decomposition.
We present a comprehensive analysis of the implications of artificial latency in the Proposer-Builder Separation framework on the Ethereum network. Focusing on the MEV-Boost auction system, we analyze how strategic latency manipulation affects Maximum Extractable Value yields and network integrity. Our findings reveal both increased profitability for node operators and significant systemic challenges, including heightened network inefficiencies and centralization risks. We empirically validates these insights with a pilot that Chorus One has been operating on Ethereum mainnet. We demonstrate the nuanced effects of latency on bid selection and validator dynamics. Ultimately, this research underscores the need for balanced strategies that optimize Maximum Extractable Value capture while preserving the Ethereum network's decentralization ethos.
Spatial regression models are central to the field of spatial statistics. Nevertheless, their estimation in case of large and irregular gridded spatial datasets presents considerable computational challenges. To tackle these computational problems, Arbia \citep{arbia_2014_pairwise} introduced a pseudo-likelihood approach (called pairwise likelihood, say PL) which required the identification of pairs of observations that are internally correlated, but mutually conditionally uncorrelated. However, while the PL estimators enjoy optimal theoretical properties, their practical implementation when dealing with data observed on irregular grids suffers from dramatic computational issues (connected with the identification of the pairs of observations) that, in most empirical cases, negatively counter-balance its advantages. In this paper we introduce an algorithm specifically designed to streamline the computation of the PL in large and irregularly gridded spatial datasets, dramatically simplifying the estimation phase. In particular, we focus on the estimation of Spatial Error models (SEM). Our proposed approach, efficiently pairs spatial couples exploiting the KD tree data structure and exploits it to derive the closed-form expressions for fast parameter approximation. To showcase the efficiency of our method, we provide an illustrative example using simulated data, demonstrating the computational advantages if compared to a full likelihood inference are not at the expenses of accuracy.
We consider the on-line coloring problem restricted to proper interval graphs with known interval representation. Chrobak and \'{S}lusarek (1981) showed that the greedy $\textrm{First-Fit}$ algorithm has a strict competitive ratio of $2$. It remains open whether there is an on-line algorithm that performs better than $\textrm{First-Fit}$. Piotr (2008) showed that if the representation is not known, there is no better on-line algorithm. Epstein and Levy (2005) showed that no on-line algorithm has a strict competitive ratio less than $1.5$ when a unit-interval representation is known, which was later improved to $1.\overline{3}$. In this paper, we show that there is no on-line algorithm with strict competitive ratio less than $1.75$ by presenting a strategy that can force any on-line algorithm to use $7$ colors on a proper interval graph $G$ with chromatic number $\chi(G)\leq 4$ and known interval representation.
Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.
We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schr\"odinger equation.
In this work we design and analyse a Discrete de Rham (DDR) method for the incompressible Navier-Stokes equations. Our focus is, more specifically, on the SDDR variant, where a reduction in the number of unknowns is obtained using serendipity techniques. The main features of the DDR approach are the support of general meshes and arbitrary approximation orders. The method we develop is based on the curl-curl formulation of the momentum equation and, through compatibility with the Helmholtz-Hodge decomposition, delivers pressure-robust error estimates for the velocity. It also enables non-standard boundary conditions, such as imposing the value of the pressure on the boundary. In-depth numerical validation on a complete panel of tests including general polyhedral meshes is provided. The paper also contains an appendix where bounds on DDR potential reconstructions and differential operators are proved in the more general framework of Polytopal Exterior Calculus.
Two important problems in the field of Topological Data Analysis are defining practical multifiltrations on objects and showing ability of TDA to detect the geometry. Motivated by the problems, we constuct three multifiltrations named multi-GENEO, multi-DGENEO and mix-GENEO, and prove the stability of both the interleaving distance and multiparameter persistence landscape of multi-GENEO with respect to the pseudometric of the subspace of bounded functions. We also give the estimations of upper bound for multi-DGENEO and mix-GENEO. Finally, we provide experiment results on MNIST dataset to demonstrate our bifiltrations have ability to detect geometric and topological differences of digital images.
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