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Journal articles on the topic 'Robust Combinatorial Optimization'

Author: Grafiati

Published: 15 June 2024

Last updated: 31 July 2025

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1

Adjiashvili, David, Sebastian Stiller, and Rico Zenklusen. "Bulk-Robust combinatorial optimization." Mathematical Programming 149, no. 1-2 (2014): 361–90. http://dx.doi.org/10.1007/s10107-014-0760-6.

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2

Kawase, Yasushi, and Hanna Sumita. "Randomized Strategies for Robust Combinatorial Optimization." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 7876–83. http://dx.doi.org/10.1609/aaai.v33i01.33017876.

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In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory and optimization. To solve the problem, we propose two types of schemes for designing approxim

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3

Buchheim, Christoph, and Jannis Kurtz. "Min–max–min robust combinatorial optimization." Mathematical Programming 163, no. 1-2 (2016): 1–23. http://dx.doi.org/10.1007/s10107-016-1053-z.

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4

Poss, Michael. "Robust combinatorial optimization with knapsack uncertainty." Discrete Optimization 27 (February 2018): 88–102. http://dx.doi.org/10.1016/j.disopt.2017.09.004.

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5

Koster, Arie M. C. A., and Michael Poss. "Special issue on: robust combinatorial optimization." EURO Journal on Computational Optimization 6, no. 3 (2018): 207–9. http://dx.doi.org/10.1007/s13675-018-0102-1.

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6

Goerigk, Marc, and Stefan Lendl. "Robust Combinatorial Optimization with Locally Budgeted Uncertainty." Open Journal of Mathematical Optimization 2 (May 18, 2021): 1–18. http://dx.doi.org/10.5802/ojmo.5.

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7

Goerigk, Marc, and Stephen J. Maher. "Generating hard instances for robust combinatorial optimization." European Journal of Operational Research 280, no. 1 (2020): 34–45. http://dx.doi.org/10.1016/j.ejor.2019.07.036.

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8

Poss, Michael. "Robust combinatorial optimization with variable cost uncertainty." European Journal of Operational Research 237, no. 3 (2014): 836–45. http://dx.doi.org/10.1016/j.ejor.2014.02.060.

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9

Dokka, Trivikram, Marc Goerigk, and Rahul Roy. "Mixed uncertainty sets for robust combinatorial optimization." Optimization Letters 14, no. 6 (2019): 1323–37. http://dx.doi.org/10.1007/s11590-019-01456-3.

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Kurtz, Jannis. "Robust combinatorial optimization under budgeted–ellipsoidal uncertainty." EURO Journal on Computational Optimization 6, no. 4 (2018): 315–37. http://dx.doi.org/10.1007/s13675-018-0097-7.

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Poss, Michael. "Robust combinatorial optimization with variable budgeted uncertainty." 4OR 11, no. 1 (2012): 75–92. http://dx.doi.org/10.1007/s10288-012-0217-9.

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12

Crema, Alejandro. "Min max min robust (relative) regret combinatorial optimization." Mathematical Methods of Operations Research 92, no. 2 (2020): 249–83. http://dx.doi.org/10.1007/s00186-020-00712-y.

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Buchheim, Christoph, and Jannis Kurtz. "Robust combinatorial optimization under convex and discrete cost uncertainty." EURO Journal on Computational Optimization 6, no. 3 (2018): 211–38. http://dx.doi.org/10.1007/s13675-018-0103-0.

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14

Chassein, André, and Marc Goerigk. "On scenario aggregation to approximate robust combinatorial optimization problems." Optimization Letters 12, no. 7 (2017): 1523–33. http://dx.doi.org/10.1007/s11590-017-1206-x.

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15

Goerigk, Marc, and Martin Hughes. "Representative scenario construction and preprocessing for robust combinatorial optimization problems." Optimization Letters 13, no. 6 (2018): 1417–31. http://dx.doi.org/10.1007/s11590-018-1348-5.

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Raith, Andrea, Marie Schmidt, Anita Schöbel, and Lisa Thom. "Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty." European Journal of Operational Research 267, no. 2 (2018): 628–42. http://dx.doi.org/10.1016/j.ejor.2017.12.018.

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17

Chassein, André, and Marc Goerigk. "Compromise solutions for robust combinatorial optimization with variable-sized uncertainty." European Journal of Operational Research 269, no. 2 (2018): 544–55. http://dx.doi.org/10.1016/j.ejor.2018.01.056.

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18

Alesiani, Francesco. "Implicit Bilevel Optimization: Differentiating through Bilevel Optimization Programming." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 12 (2023): 14683–91. http://dx.doi.org/10.1609/aaai.v37i12.26716.

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Bilevel Optimization Programming is used to model complex and conflicting interactions between agents, for example in Robust AI or Privacy preserving AI. Integrating bilevel mathematical programming within deep learning is thus an essential objective for the Machine Learning community. Previously proposed approaches only consider single-level programming. In this paper, we extend existing single-level optimization programming approaches and thus propose Differentiating through Bilevel Optimization Programming (BiGrad) for end-to-end learning of models that use Bilevel Programming as a layer. B

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19

Montajabiha, Mahsa, Alireza Arshadi Khamseh, and Behrouz Afshar-Nadjafi. "A robust algorithm for project portfolio selection problem using real options valuation." International Journal of Managing Projects in Business 10, no. 2 (2017): 386–403. http://dx.doi.org/10.1108/ijmpb-12-2015-0114.

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Purpose The principal concern of organization managers in the global rivalry of commerce environment is how to select the project portfolio among available projects. In this matter, organizations should consider the uncertainty intrinsic in the projects regarding an appropriate valuation technique within an optimization framework. In this research, the purpose of this paper is to formulate using a robust optimization algorithm to deal with the complexities and uncertainty inherent in the construction of the project portfolio. Design/methodology/approach First, a general mathematical formulatio

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20

Shahsavar, Moslem, Amir Abbas Najafi, and Seyed Taghi Akhavan Niaki. "Statistical Design of Genetic Algorithms for Combinatorial Optimization Problems." Mathematical Problems in Engineering 2011 (2011): 1–17. http://dx.doi.org/10.1155/2011/872415.

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Many genetic algorithms (GA) have been applied to solve different NP-complete combinatorial optimization problems so far. The striking point of using GA refers to selecting a combination of appropriate patterns in crossover, mutation, and and so forth and fine tuning of some parameters such as crossover probability, mutation probability, and and so forth. One way to design a robust GA is to select an optimal pattern and then to search for its parameter values using a tuning procedure. This paper addresses a methodology to both optimal pattern selection and the tuning phases by taking advantage

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21

Zhang , Tianyi, and Yuan Ke. "\({\ell_0}\) Optimization with Robust Non-Oracular Quantum Search." Technologies 11, no. 5 (2023): 148. http://dx.doi.org/10.3390/technologies11050148.

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In this article, we introduce an innovative hybrid quantum search algorithm, the Robust Non-oracle Quantum Search (RNQS), which is specifically designed to efficiently identify the minimum value within a large set of random numbers. Distinct from the Grover’s algorithm, the proposed RNQS algorithm circumvents the need for an oracle function that describes the true solution state, a feature often impractical for data science applications. Building on existing non-oracular quantum search algorithms, RNQS enhances robustness while substantially reducing running time. The superior properties of RN

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22

Omer, Jérémy, Michael Poss, and Maxime Rougier. "Combinatorial Robust Optimization with Decision-Dependent Information Discovery and Polyhedral Uncertainty." Open Journal of Mathematical Optimization 5 (September 5, 2024): 1–25. http://dx.doi.org/10.5802/ojmo.33.

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23

Pessoa, Artur Alves, Michael Poss, Ruslan Sadykov, and François Vanderbeck. "Branch-Cut-and-Price for the Robust Capacitated Vehicle Routing Problem with Knapsack Uncertainty." Operations Research 69, no. 3 (2021): 739–54. http://dx.doi.org/10.1287/opre.2020.2035.

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Capacitated vehicle routing problems are widely studied combinatorial optimization problems, and branch-and-cut-and-price algorithms can solve instances harder than ever before. These models, however, neglect that demands volumes are often not known with precision when planning the vehicle routes, thus incentivizing decision makers to significantly overestimate the volumes for avoiding coping with infeasible routes. A robust formulation that models demand uncertainty through a knapsack polytope is considered. A new branch-and-cut-and-price algorithm for the problem is provided, which combines

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24

Buchheim, Christoph, and Marianna De Santis. "An active set algorithm for robust combinatorial optimization based on separation oracles." Mathematical Programming Computation 11, no. 4 (2019): 755–89. http://dx.doi.org/10.1007/s12532-019-00160-8.

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25

Yaman, Hande. "Short Paper - A Note on Robust Combinatorial Optimization with Generalized Interval Uncertainty." Open Journal of Mathematical Optimization 4 (June 5, 2023): 1–7. http://dx.doi.org/10.5802/ojmo.23.

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26

Lakhdari, Saliha, and Fateh Boutekkouk. "Optimization Trends for Wireless Network On-Chip." International Journal of Wireless Networks and Broadband Technologies 10, no. 1 (2021): 1–31. http://dx.doi.org/10.4018/ijwnbt.2021010101.

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Designing sustainable and high-performance wireless multi-core chips requires a matchless tradeoff between many aspects including scalable and reliable architectures implementation which in its turn implies aware-wideband energy-efficient wireless interfaces and adopting innovative straightforward optimization approaches to achieve the optimal configuration with a minimal cost. This paper focuses on investigating various existing designs and methodologies for wireless network on chip (WiNoC) architectures, as well as the different emerging technologies and optimization tools for the design of

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27

McElfresh, Duncan C., Hoda Bidkhori, and John P. Dickerson. "Scalable Robust Kidney Exchange." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 1077–84. http://dx.doi.org/10.1609/aaai.v33i01.33011077.

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In barter exchanges, participants directly trade their endowed goods in a constrained economic setting without money. Transactions in barter exchanges are often facilitated via a central clearinghouse that must match participants even in the face of uncertainty—over participants, existence and quality of potential trades, and so on. Leveraging robust combinatorial optimization techniques, we address uncertainty in kidney exchange, a real-world barter market where patients swap (in)compatible paired donors. We provide two scalable robust methods to handle two distinct types of uncertainty in ki

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28

Skybytskyi, N. M. "Robust time to buy and sell stock." Journal of Numerical and Applied Mathematics, no. 1 (2025): 90–100. https://doi.org/10.17721/2706-9699.2025.1.08.

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This paper focuses on the robust version of the classical algorithmic problem of finding the best time to buy and sell stock. We consider its variants with different transaction limits, as well as other modifications like transaction fee or cooldown. We reduce each classical problem to its robust version, thereby obtaining lower bounds on the time complexity of all potential solutions. We extensively test all developed methods on random and adversarial data to ensure correctness and evaluate performance. We propose efficient methods for the robust counterparts of almost all problems. We also d

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29

Buchheim, Christoph. "A note on the nonexistence of oracle-polynomial algorithms for robust combinatorial optimization." Discrete Applied Mathematics 285 (October 2020): 591–93. http://dx.doi.org/10.1016/j.dam.2020.07.002.

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30

Álvarez-Miranda, Eduardo, Ivana Ljubić, and Paolo Toth. "A note on the Bertsimas & Sim algorithm for robust combinatorial optimization problems." 4OR 11, no. 4 (2013): 349–60. http://dx.doi.org/10.1007/s10288-013-0231-6.

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31

Lee, Taehan, and Changhyun Kwon. "A short note on the robust combinatorial optimization problems with cardinality constrained uncertainty." 4OR 12, no. 4 (2014): 373–78. http://dx.doi.org/10.1007/s10288-014-0270-7.

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32

Huang, Changlong, Ling Pei, and Qi Ouyang. "Research on the Cuckoo Algorithm for Flexible Workshop Scheduling Problems." Frontiers in Computing and Intelligent Systems 3, no. 1 (2023): 177–81. http://dx.doi.org/10.54097/fcis.v3i1.6365.

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As a typical combinatorial optimization problem, the essence of the solution to the scheduling problem is to formulate a reasonable scheduling scheme to arrange and allocate production resources, thereby obtaining the optimal scheduling result. The Cuckoo Search Algorithm (CS), a revolutionary meta-heuristic, is based on the cuckoo's breeding behavior and combines Lévy flight and random walk strategies to more efficiently attain the search goal. CS algorithm has been extensively employed in a variety of intricate combinatorial optimization issues, with its few parameters, precise resolution, r

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33

Patish, Uri, and Shimon Ullman. "Cakewalk Sampling." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 03 (2020): 2400–2407. http://dx.doi.org/10.1609/aaai.v34i03.5620.

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We study the task of finding good local optima in combinatorial optimization problems. Although combinatorial optimization is NP-hard in general, locally optimal solutions are frequently used in practice. Local search methods however typically converge to a limited set of optima that depend on their initialization. Sampling methods on the other hand can access any valid solution, and thus can be used either directly or alongside methods of the former type as a way for finding good local optima. Since the effectiveness of this strategy depends on the sampling distribution, we derive a robust le

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34

Montoya, Oscar Danilo, Walter Gil-González, and Jesus C. Hernández. "Optimal Scheduling of Photovoltaic Generators in Asymmetric Bipolar DC Grids Using a Robust Recursive Quadratic Convex Approximation." Machines 11, no. 2 (2023): 177. http://dx.doi.org/10.3390/machines11020177.

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This paper presents a robust quadratic convex model for the optimal scheduling of photovoltaic generators in unbalanced bipolar DC grids. The proposed model is based on Taylor’s series expansion which relaxes the hyperbolic relation between constant power terminals and voltage profiles. Furthermore, the proposed model is solved in the recursive form to reduce the error generated by relaxations assumed. Additionally, uncertainties in PV generators are considered to assess the effectiveness of the proposed recursive convex. Several proposed scenarios for the numerical validations in a modified 2

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35

Elhenawy, Mohammed, Ahmad Abutahoun, Taqwa I. Alhadidi, et al. "Visual Reasoning and Multi-Agent Approach in Multimodal Large Language Models (MLLMs): Solving TSP and mTSP Combinatorial Challenges." Machine Learning and Knowledge Extraction 6, no. 3 (2024): 1894–921. http://dx.doi.org/10.3390/make6030093.

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Multimodal Large Language Models (MLLMs) harness comprehensive knowledge spanning text, images, and audio to adeptly tackle complex problems. This study explores the ability of MLLMs in visually solving the Traveling Salesman Problem (TSP) and Multiple Traveling Salesman Problem (mTSP) using images that portray point distributions on a two-dimensional plane. We introduce a novel approach employing multiple specialized agents within the MLLM framework, each dedicated to optimizing solutions for these combinatorial challenges. We benchmarked our multi-agent model solutions against the Google OR

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36

Conde, Eduardo. "Robust minmax regret combinatorial optimization problems with a resource–dependent uncertainty polyhedron of scenarios." Computers & Operations Research 103 (March 2019): 97–108. http://dx.doi.org/10.1016/j.cor.2018.10.014.

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37

Büsing, Christina, and Sabrina Schmitz. "Robust two-stage combinatorial optimization problems under discrete demand uncertainties and consistent selection constraints." Discrete Applied Mathematics 347 (April 2024): 187–213. http://dx.doi.org/10.1016/j.dam.2023.12.028.

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38

Gao, Zhichao, Mingfa Zheng, Haitao Zhong, and Yu Mei. "Robust Optimization for Cooperative Task Assignment of Heterogeneous Unmanned Aerial Vehicles with Time Window Constraints." Axioms 14, no. 3 (2025): 184. https://doi.org/10.3390/axioms14030184.

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The cooperative task assignment problem with time windows for heterogeneous multiple unmanned aerial vehicles is an attractive complex combinatorial optimization problem. In reality, unmanned aerial vehicles’ fuel consumption exhibits uncertainty due to environmental factors or operational maneuvers, and accurately determining the probability distributions for these uncertainties remains challenging. This paper investigates the heterogeneous multiple unmanned aerial vehicle cooperative task assignment model that incorporates time window constraints under uncertain environments. To model the ti

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39

Sugiarto. "A Hybrid Optimization Algorithm for Large-Scale Combinatorial Problems in Cloud Computing Environments." ALCOM: Journal of Algorithm and Computing 1, no. 1 (2025): 1–12. https://doi.org/10.63846/v3km6g84.

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Combinatorial problems, such as task scheduling and resource allocation, present significant challenges in cloud computing due to the exponential growth of solution spaces. Conventional optimization algorithms often prove inadequate in efficiently handling large-scale problems, resulting in suboptimal resource utilization and increased operational costs. To address these limitations, this study proposes a hybrid optimization algorithm that combines the exploration capabilities of metaheuristic methods with the precision of exact optimization techniques. The proposed approach integrates genetic

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40

Coloma-Salazar, Manuel-Enrique, José Arzola-Ruiz, Clara-Elena Marrero-Fornaris, Vladimir Socha, and Umer Asgher. "A Combinatorial Approach for Optimizing Transportation System: Multi-Objective Decision-Making Framework." Neural Network World 34, no. 3 (2024): 135–68. https://doi.org/10.14311/nnw.2024.34.008.

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This study presents a comprehensive multi-objective transportation model aimed at optimizing complex vehicle routing problems, which are nondeterministic polynomial time NP-hard due to spatial, temporal, and capacity constraints. In this study, the multi-objective transportation model integrates decisionmaker preferences with hybrid optimization techniques, including the approximatecombinatorial method, ant colony optimization and evolutionary algorithms. it seeks to minimize transportation costs, time, and emissions while accounting for real-world constraints such as fleet composition, custom

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41

Wang, Xinggang, Zhengdong Zhang, Yi Ma, Xiang Bai, Wenyu Liu, and Zhuowen Tu. "Robust Subspace Discovery via Relaxed Rank Minimization." Neural Computation 26, no. 3 (2014): 611–35. http://dx.doi.org/10.1162/neco_a_00555.

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This letter examines the problem of robust subspace discovery from input data samples (instances) in the presence of overwhelming outliers and corruptions. A typical example is the case where we are given a set of images; each image contains, for example, a face at an unknown location of an unknown size; our goal is to identify or detect the face in the image and simultaneously learn its model. We employ a simple generative subspace model and propose a new formulation to simultaneously infer the label information and learn the model using low-rank optimization. Solving this problem enables us

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42

Climent, Laura, Richard J. Wallace, Barry O'Sullivan, and Eugene C. Freuder. "Extrapolating from Limited Uncertain Information in Large-Scale Combinatorial Optimization Problems to Obtain Robust Solutions." International Journal on Artificial Intelligence Tools 25, no. 01 (2016): 1660005. http://dx.doi.org/10.1142/s0218213016600058.

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Data uncertainty in real-life problems is a current challenge in many areas, including Operations Research (OR) and Constraint Programming (CP). This is especially true given the continual and accelerating increase in the amount of data associated with real-life problems, to which Large Scale Combinatorial Optimization (LSCO) techniques may be applied. Although data uncertainty has been studied extensively in the literature, many approaches do not take into account the partial or complete lack of information about uncertainty in real-life settings. To meet this challenge, in this paper we pres

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43

Wang, Fengmin, Dachuan Xu, and Chenchen Wu. "Combinatorial approximation algorithms for the robust facility location problem with penalties." Journal of Global Optimization 64, no. 3 (2014): 483–96. http://dx.doi.org/10.1007/s10898-014-0251-6.

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44

Sagban, Rafid, Ku Ruhana Ku-Mahamud, and Muhamad Shahbani Abu Bakar. "ACOustic: A Nature-Inspired Exploration Indicator for Ant Colony Optimization." Scientific World Journal 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/392345.

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A statistical machine learning indicator,ACOustic, is proposed to evaluate the exploration behavior in the iterations of ant colony optimization algorithms. This idea is inspired by the behavior of some parasites in their mimicry to the queens’ acoustics of their ant hosts. The parasites’ reaction results from their ability to indicate the state of penetration. The proposed indicator solves the problem of robustness that results from the difference of magnitudes in the distance’s matrix, especially when combinatorial optimization problems with rugged fitness landscape are applied. The performa

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45

Lazzarinetti, Giorgio, Riccardo Dondi, Sara Manzoni, and Italo Zoppis. "DLMinTC+: A Deep Learning Based Algorithm for Minimum Timeline Cover on Temporal Graphs." Algorithms 18, no. 2 (2025): 113. https://doi.org/10.3390/a18020113.

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Combinatorial optimization on temporal graphs is critical for summarizing dynamic networks in various fields, including transportation, social networks, and biology. Among these problems, the Minimum Timeline Cover (MinTCover) problem, aimed at identifying minimal activity intervals for representing temporal interactions, remains underexplored in the context of advanced machine learning techniques. Existing heuristic and approximate methods, while effective in certain scenarios, struggle with capturing complex temporal dependencies and scalability in dense, large-scale networks. Addressing thi

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Zhou, Tao, Liang Luo, Shengchen Ji, and Yuanxin He. "A Reinforcement Learning Approach to Robust Scheduling of Permutation Flow Shop." Biomimetics 8, no. 6 (2023): 478. http://dx.doi.org/10.3390/biomimetics8060478.

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The permutation flow shop scheduling problem (PFSP) stands as a classic conundrum within the realm of combinatorial optimization, serving as a prevalent organizational structure in authentic production settings. Given that conventional scheduling approaches fall short of effectively addressing the intricate and ever-shifting production landscape of PFSP, this study proposes an end-to-end deep reinforcement learning methodology with the objective of minimizing the maximum completion time. To tackle PFSP, we initially model it as a Markov decision process, delineating pertinent states, actions,

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Lin, Shanxian, Yifei Yang, Yuichi Nagata, and Haichuan Yang. "Elite Evolutionary Discrete Particle Swarm Optimization for Recommendation Systems." Mathematics 13, no. 9 (2025): 1398. https://doi.org/10.3390/math13091398.

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Recommendation systems (RSs) play a vital role in e-commerce and content platforms, yet balancing efficiency and recommendation quality remains challenging. Traditional deep models are computationally expensive, while heuristic methods like particle swarm optimization struggle with discrete optimization. To address these limitations, this paper proposes elite-evolution-based discrete particle swarm optimization (EEDPSO), a novel framework specifically designed to optimize high-dimensional combinatorial recommendation tasks. EEDPSO restructures the velocity and position update mechanisms to ope

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48

Kushwaha, Ajay Shriram. "Bio-inspired Algorithms for Optimization in Mechanical Design Systems." International Journal of Research in Modern Engineering & Emerging Technology 10, no. 12 (2022): 1–9. https://doi.org/10.63345/ijrmeet.org.v10.i12.1.

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Bio-inspired algorithms have emerged as powerful optimization techniques, offering robust, adaptive, and scalable solutions for complex engineering design problems. This manuscript investigates the application of four representative bio-inspired algorithms—Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and Artificial Bee Colony (ABC)—to optimization challenges in mechanical design systems up to the year 2022. We present a detailed comparison of their performance on benchmark design tasks such as truss sizing, cam profile optimization, and mechanism pa

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49

Gupta, Ankit, Anand Kumar, Ishrat Ali, Geetanjali ., and Uma Shankar. "Enhancement of Computer Network Communication Security Encryption System Based on Ant colony optimization Algorithm." International Research Journal of Computer Science 11, no. 10 (2024): 609–12. https://doi.org/10.26562/irjcs.2024.v1110.03.

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In the digital era, the exponential growth of computer networks has heightened the need for robust communication security to protect sensitive data from breaches and cyber attacks. This research explores the enhancement of network communication security by developing an encryption system inspired by the Ant Colony Optimization (ACO) algorithm. ACO, a bio-inspired met heuristic, is particularly effective in solving combinatorial optimization problems, making it well-suited for the dynamic and complex nature of cryptographic key generation and data encryption. This research contributes to the fi

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50

Zhou, Guangyan, and Rui Kang. "On the Lower Bounds of (1,0)-Super Solutions for Random k-SAT." International Journal of Foundations of Computer Science 30, no. 02 (2019): 247–54. http://dx.doi.org/10.1142/s0129054119500035.

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Super solution is a notion introduced to produce robust and stable solutions of combinatorial optimization and decision problems. We consider the [Formula: see text]-super solutions of random instances of [Formula: see text]-SAT, where a clause is satisfied if and only if there are at least two satisfied literals in this clause. By using an enhanced weighting scheme, we obtain better lower bounds that, if a random [Formula: see text]-CNF formula [Formula: see text] is [Formula: see text]-unsatisfiable with probability tending to 1 as [Formula: see text], then [Formula: see text] for [Formula:

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