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On the Locality of the Lovász Local Lemma

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arXiv 2025年

作者： Davies-Peck, Peter Durham University United Kingdom

The Lovász Local Lemma is a versatile result in probability theory, characterizing circumstances in which a collection of n ‘bad events’, each occurring with probability at most p and dependent on a set of underlying random variables, can be avoided. It is a central tool of the probabilistic method, since it can be used to show that combinatorial objects satisfying some desirable properties must exist. While the original proof was existential, subsequent work has shown algorithms for the Lovász Local Lemma: that is, in circumstances in which the lemma proves the existence of some object, these algorithms can constructively find such an object. One main strand of these algorithms, which began with Moser and Tardos’s well-known result (JACM 2010), involves iteratively resampling the dependent variables of satisfied bad events until none remain satisfied. In this paper, we present a novel analysis that can be applied to resampling-style Lovász Local Lemma algorithms. This analysis shows that an output assignment for the dependent variables of most events can be determined only from O(log log1/p n)-radius local neighborhoods, and that the events whose variables may still require resampling can be identified from these neighborhoods. This allows us to improve randomized complexities for the constructive Lovász Local Lemma (with polynomial criterion) in several parallel and distributed models. In particular, we obtain: • A LOCAL algorithm with O(log log1/p n) node-averaged complexity (while matching the O(log1/p n) worst-case complexity of Chung, Pettie, and Su). • An algorithm for the LCA and VOLUME models requiring dO(log log1/p n) probes per query. • An O(log log log1/p n)-round algorithm for CONGESTED CLIQUE, linear space MPC, and Heterogenous MPC. © 2025, CC BY.

关键词： consensus algorithm

A fast algorithm for the Frobenius problem in three variables

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arXiv 2025年

作者： Rosin, Daniel

Given the function Γ(X) = a1x1 + a2x2 + a3x3, where the set {a1, a2, a3}, denoted A, consists of positive integers and the set {x1, x2, x3}, denoted X, consists of non-negative integers, the Frobenius problem in three variables is to find the greatest integer, which is not in the codomain of Γ. The fastest known algorithm for solving the three variable case of the Frobenius problem was invented by H. Greenberg in 1988 whose worst case time complexity is a logarithmic function of A. In 2017 A. Tripathi presented another algorithm for solving the same problem. This article presents an algorithm whose foundation is the same as Tripathi’s. However, this algorithm is significantly different from Tripathi’s and we show that its worst case time complexity is also a logarithmic function of *** Codes 11D07 © 2025, CC BY.

关键词： consensus algorithm

Improved Space Bounds for Subset Sum

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arXiv 2024年

作者： Belova, Tatiana Chukhin, Nikolai Kulikov, Alexander S. Mihajlin, Ivan St. Petersburg Department of Steklov Mathematical Institute Russian Academy of Sciences Russia Neapolis University Pafos JetBrains Research Cyprus JetBrains Research Czech Republic

More than 40 years ago, Schroeppel and Shamir presented an algorithm that solves the Subset Sum problem for n integers in time O∗(20.5n) and space O∗(20.25n). The time upper bound remains unbeaten, but the space upper bound has been improved to O∗(20.249999n) in a recent breakthrough paper by Nederlof and Węgrzycki (STOC 2021). Their algorithm is a clever combination of a number of previously known techniques with a new reduction and a new algorithm for the Orthogonal Vectors problem. In this paper, we give two new algorithms for Subset Sum. We start by presenting an Arthur-Merlin algorithm: upon receiving the verifier's randomness, the prover sends an n/4-bit long proof to the verifier who checks it in (deterministic) time and space O∗(2n/4). An interesting consequence of this result is the following fine-grained lower bound: assuming that 4-SUM cannot be solved in time O(n2−Ε) for all Ε > 0, Circuit SAT cannot be solved in time O(g2(1-Ε)n), for all Ε > 0 (where n and g denote the number of inputs and the number of gates, respectively). Then, we improve the space bound by Nederlof and Węgrzycki to O∗(20.246n) and also simplify their algorithm and its analysis. We achieve this space bound by further filtering sets of subsets using a random prime number. This allows us to reduce an instance of Subset Sum to a larger number of instances of smaller size. © 2024, CC BY.

关键词： consensus algorithm

Advanced Veterinary Clinic Management Predictive Analytics and Blockchain Innovation

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Advanced Veterinary Clinic Management Predictive Analytics a...

Applied Artificial Intelligence and Computing (ICAAIC), International Conference on

作者： R. Vijay Sai B. G. Geetha A Hariharan A Haripriya K Kameshwaran Computer Science and Engineering K S Rangasamy College of Technology Tamil Nadu India

ISBN: (数字)9798350375190

ISBN: (纸本)9798350375206

Enhancing the security, effectiveness, and transparency of healthcare data, distributed ledger technology like blockchain holds the potential to completely transform the healthcare sector. Sensitive patient information is more difficult for bad actors to bre ach thanks to the SHA-256 algorithm, which improves data security. Standardized data formats and smart contracts make it easier for disparate healthcare systems to communicate with one another. Block chain deployment in the healthcare sector is not without its challenges, though, including user uptake, scalability, and regulatory compliance. By highlighting the potential benefits and difficulties of incorporating block chain technology in healthcare management systems, this study provides significant insights into the evolving landscape of healthcare data management.

关键词： Technological innovation Scalability Smart contracts Medical services consensus algorithm Transforms Proof of Work

Pareto Set Identification With Posterior Sampling

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arXiv 2024年

作者： Kone, Cyrille Jourdan, Marc Kaufmann, Emilie Univ. Lille CNRS Inria Centrale Lille UMR 9189-CRIStAL LilleF-59000 France EPFL Lausanne Switzerland

The problem of identifying the best answer among a collection of items having real-valued distribution is well-understood. Despite its practical relevance for many applications, fewer works have studied its extension when multiple and potentially conflicting metrics are available to assess an item’s quality. Pareto set identification (PSI) aims to identify the set of answers whose means are not uniformly worse than another. This paper studies PSI in the transductive linear setting with potentially correlated objectives. Building on posterior sampling in both the stopping and the sampling rules, we propose the PSIPS algorithm that deals simultaneously with structure and correlation without paying the computational cost of existing oracle-based algorithms. Both from a frequentist and Bayesian perspective, PSIPS is asymptotically optimal. We demonstrate its good empirical performance in real-world and synthetic instances. © 2024, CC BY.

关键词： consensus algorithm

Sliding-Window Thompson Sampling for Non-Stationary Settings

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arXiv 2024年

作者： Fiandri, Marco Metelli, Alberto Maria Trovò, Francesco Politecnico di Milano Italy

Restless Bandits describe sequential decision-making problems in which the rewards evolve with time independently from the actions taken by the policy-maker. It has been shown that classical Bandit algorithms fail when the underlying environment is changing, making clear that in order to tackle more challenging scenarios specifically crafted algorithms are needed. In this paper, extending and correcting the work by Trovò et al. [2020], we analyze two Thompson-Sampling inspired algorithms, namely BETA-SWTS and γ-SWGTS, introduced to face the additional complexity given by the non-stationary nature of the settings;in particular we derive a general formulation for the regret in any arbitrary restless environment for both Bernoulli and Subgaussian rewards, and, through the introduction of new quantities, we delve in what contribution lays the deeper foundations of the error made by the algorithms. Finally, we infer from the general formulation the regret for two of the most common non-stationary settings: the Abruptly Changing and the Smoothly Changing environments. © 2024, CC BY.

关键词： consensus algorithm

Generalising the maximum independent set algorithm via Boolean networks

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arXiv 2024年

作者： Gadouleau, Maximilien Kutner, David C. Department of Computer Science Durham University Durham United Kingdom

A simple greedy algorithm to find a maximal independent set (MIS) in a graph starts with the empty set and visits every vertex, adding it to the set if and only if none of its neighbours are already in the set. In this paper, we consider the generalisation of this so-called MIS algorithm by allowing it to start with any set of vertices and we prove the hardness of many decision problems related to this generalisation. Our results are based on two main strategies. Firstly, we view the MIS algorithm as a sequential update of a Boolean network, which we shall refer to as the MIS network, according to a permutation of the vertex set. The set of fixed points of the MIS network corresponds to the set of MIS of the graph. Our generalisation then consists in allowing to start from any configuration and to follow a sequential update given by a word of vertices. Secondly, we introduce the concept of a constituency of a graph, that is a set of vertices that is dominated by an independent set. Deciding whether a set of vertices is a constituency is NP-complete;decision problems related to the MIS algorithm will be reduced from the Constituency problem or its variants. In this paper, we first consider universal configurations, i.e. those that can reach all maximal independent sets;deciding whether a configuration is universal is coNP-complete. Second, we consider so-called fixing words, that allow to reach a MIS regardless of the starting configuration, and fixing permutations, which we call permises;deciding whether a permutation is fixing is coNP-complete. Third, we consider permissible graphs, i.e. those graphs that have a permis. We construct large classes of permissible and non-permissible graphs, notably near-comparability graphs which may be of independent interest;deciding whether a graph is permissible is coNP-hard. Finally, we generalise the MIS algorithm to digraphs. In this case, the algorithm uses the so-called kernel network, whose fixed points are the kernels of t

关键词： consensus algorithm

Computing the extinction path for epidemic models

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arXiv 2024年

作者： Clancy, Damian Stewart, John J.H. Department of Actuarial Mathematics and Statistics Maxwell Institute for Mathematical Sciences Heriot-Watt University EdinburghEH14 4AS United Kingdom

In infectious disease modelling, the expected time from endemicity to extinction (of infection) may be analysed via WKB approximation, a method with origins in mathematical physics. The method is very general, but its uptake to date may have been limited by the practical difficulties of implementation. It is necessary to compute a trajectory of a (high dimensional) dynamical system, the ‘extinction path’, and this trajectory is maximally sensitive to small perturbations, making numerical computation challenging. Our objective here is to make this methodology more accessible by presenting four computational algorithms, with associated Matlab code, together with discussion of various ways in which the algorithms may be tuned to achieve satisfactory convergence. We illustrate our methods using three standard infectious disease models. For each such model, we demonstrate that our algorithms are able to improve upon previously available results. Copyright © 2024, The Authors. All rights reserved.

关键词： consensus algorithm

Scheduling With Time Discounts

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arXiv 2024年

作者： Gafni, Yotam Yaish, Aviv Weizmann Institute Yale University United States

We study a financial version of the classic online problem of scheduling weighted packets with deadlines. The main novelty is that, while previous works assume packets have fixed weights throughout their lifetime, this work considers packets with time-decaying values. Such considerations naturally arise and have wide applications in financial environments, where the present value of future actions may be discounted. We analyze the competitive ratio guarantees of scheduling algorithms under a range of discount rates encompassing the "traditional" undiscounted case where weights are fixed (i.e., a discount rate of 1), the fully discounted "myopic" case (i.e., a rate of 0), and those in between. We show how existing methods from the literature perform suboptimally in the more general discounted setting. Notably, we devise a novel memoryless deterministic algorithm, and prove that it guarantees the best possible competitive ratio attainable by deterministic algorithms for discount factors up to ≈ 0.77. Moreover, we develop a randomized algorithm and prove that it outperforms the best possible deterministic algorithm, for any discount rate. While we highlight the relevance of our framework and results to blockchain transaction scheduling in particular, our approach and analysis techniques are general and may be of independent interest. © 2024, CC BY.

关键词： consensus algorithm