{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T19:35:20Z","timestamp":1787168120976,"version":"build-2736575974"},"reference-count":57,"publisher":"Public Library of Science (PLoS)","issue":"12","license":[{"start":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T00:00:00Z","timestamp":1735516800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS ONE"],"abstract":"<jats:p>The objective of the max-cut problem is to cut any graph in such a way that the total weight of the edges that are cut off is maximum in both subsets of vertices that are divided due to the cut of the edges. Although it is an elementary graph partitioning problem, it is one of the most challenging combinatorial optimization-based problems, and tons of application areas make this problem highly admissible. Due to its admissibility, the problem is solved using the Harris Hawk Optimization algorithm (HHO). Though HHO effectively solved some engineering optimization problems, is sensitive to parameter settings and may converge slowly, potentially getting trapped in local optima. Thus, HHO and some additional operators are used to solve the max-cut problem. Crossover and refinement operators are used to modify the fitness of the hawk in such a way that they can provide precise results. A mutation mechanism along with an adjustment operator has improvised the outcome obtained from the updated hawk. To accept the potential result, the acceptance criterion has been used, and then the repair operator is applied in the proposed approach. The proposed system provided comparatively better outcomes on the G-set dataset than other state-of-the-art algorithms. It obtained 533 cuts more than the discrete cuckoo search algorithm in 9 instances, 1036 cuts more than PSO-EDA in 14 instances, and 1021 cuts more than TSHEA in 9 instances. But for four instances, the cuts are lower than PSO-EDA and TSHEA. Besides, the statistical significance has also been tested using the Wilcoxon signed rank test to provide proof of the superior performance of the proposed method. In terms of solution quality, MC-HHO can produce outcomes that are quite competitive when compared to other related state-of-the-art algorithms.<\/jats:p>","DOI":"10.1371\/journal.pone.0315842","type":"journal-article","created":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T13:27:36Z","timestamp":1735565256000},"page":"e0315842","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":7,"title":["Solving the maximum cut problem using Harris Hawk Optimization algorithm"],"prefix":"10.1371","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0777-6747","authenticated-orcid":true,"given":"Md. Rafiqul","family":"Islam","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1159-8688","authenticated-orcid":true,"given":"Md. Shahidul","family":"Islam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pritam Khan","family":"Boni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aldrin Saurov","family":"Sarker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Md. Asif","family":"Anam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2024,12,30]]},"reference":[{"key":"pone.0315842.ref001","unstructured":"Wikipedia. Maximum cut; accessed on 24 March 2023. https:\/\/en.wikipedia.org\/wiki\/Maximum_cut."},{"key":"pone.0315842.ref002","unstructured":"Wheeler JW. An investigation of the max-cut problem. University of Iowa. 2004;."},{"issue":"3","key":"pone.0315842.ref003","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1287\/opre.36.3.493","article-title":"An application of combinatorial optimization to statistical physics and circuit layout design","volume":"36","author":"F Barahona","year":"1988","journal-title":"Operations Research"},{"key":"pone.0315842.ref004","doi-asserted-by":"crossref","unstructured":"Poland J, Zeugmann T. Clustering pairwise distances with missing data: Maximum cuts versus normalized cuts. In: International Conference on Discovery Science. Springer; 2006. p. 197\u2013208.","DOI":"10.1007\/11893318_21"},{"issue":"2","key":"pone.0315842.ref005","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1016\/j.cor.2007.10.001","article-title":"Hybridizing the cross-entropy method: An application to the max-cut problem","volume":"36","author":"M Laguna","year":"2009","journal-title":"Computers & Operations Research"},{"key":"pone.0315842.ref006","doi-asserted-by":"crossref","unstructured":"Fortez G, Robledo F, Romero P, Viera O. A fast genetic algorithm for the max cut-clique problem. In: International Conference on Machine Learning, Optimization, and Data Science. Springer; 2020. p. 528\u2013539.","DOI":"10.1007\/978-3-030-64583-0_47"},{"key":"pone.0315842.ref007","doi-asserted-by":"crossref","unstructured":"Sen S. Simulated annealing approach to the max cut problem. In: Applications of Artificial Intelligence 1993: Knowledge-Based Systems in Aerospace and Industry. vol. 1963. SPIE; 1993. p. 61\u201366.","DOI":"10.1117\/12.141755"},{"issue":"1","key":"pone.0315842.ref008","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1287\/ijoc.1080.0275","article-title":"Advanced scatter search for the max-cut problem","volume":"21","author":"R Mart\u00ed","year":"2009","journal-title":"INFORMS Journal on Computing"},{"key":"pone.0315842.ref009","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/s10732-011-9189-8","article-title":"Solving large scale max cut problems via tabu search","volume":"19","author":"GA Kochenberger","year":"2013","journal-title":"Journal of Heuristics"},{"issue":"2","key":"pone.0315842.ref010","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1137\/S1052623400382467","article-title":"Rank-two relaxation heuristics for max-cut and other binary quadratic programs","volume":"12","author":"S Burer","year":"2002","journal-title":"SIAM Journal on Optimization"},{"issue":"2","key":"pone.0315842.ref011","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.ejor.2008.07.026","article-title":"A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems","volume":"197","author":"AF Ling","year":"2009","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"pone.0315842.ref012","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/0167-6377(81)90020-1","article-title":"Weakly bipartite graphs and the max-cut problem","volume":"1","author":"M Gr\u00f6tschel","year":"1981","journal-title":"Operations research letters"},{"issue":"6","key":"pone.0315842.ref013","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1080\/1055678021000090033","article-title":"Randomized heuristics for the MAX-CUT problem","volume":"17","author":"P Festa","year":"2002","journal-title":"Optimization methods and software"},{"issue":"1-3","key":"pone.0315842.ref014","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/0166-218X(94)00155-7","article-title":"Solving the max-cut problem using eigenvalues","volume":"62","author":"S Poljak","year":"1995","journal-title":"Discrete Applied Mathematics"},{"key":"pone.0315842.ref015","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1016\/j.asoc.2015.04.033","article-title":"A tabu search based hybrid evolutionary algorithm for the max-cut problem","volume":"34","author":"Q Wu","year":"2015","journal-title":"Applied Soft Computing"},{"key":"pone.0315842.ref016","doi-asserted-by":"crossref","unstructured":"Wu Q, Hao JK. A memetic approach for the max-cut problem. In: Parallel Problem Solving from Nature-PPSN XII: 12th International Conference, Taormina, Italy, September 1-5, 2012, Proceedings, Part II 12. Springer; 2012. p. 297\u2013306.","DOI":"10.1007\/978-3-642-32964-7_30"},{"issue":"1","key":"pone.0315842.ref017","first-page":"3420671","article-title":"An Integrated Method Based on PSO and EDA for the Max-Cut Problem","volume":"2016","author":"G Lin","year":"2016","journal-title":"Computational intelligence and neuroscience"},{"key":"pone.0315842.ref018","doi-asserted-by":"crossref","unstructured":"Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings of ICNN\u201995-international conference on neural networks. vol. 4. ieee; 1995. p. 1942\u20131948.","DOI":"10.1109\/ICNN.1995.488968"},{"key":"pone.0315842.ref019","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s11047-007-9050-z","article-title":"A review of particle swarm optimization. Part II: hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications","volume":"7","author":"A Banks","year":"2008","journal-title":"Natural Computing"},{"key":"pone.0315842.ref020","unstructured":"Kim SH, Kim YH, Moon BR. A hybrid genetic algorithm for the max cut problem. In: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation; 2001. p. 416\u2013423."},{"key":"pone.0315842.ref021","doi-asserted-by":"crossref","first-page":"113428","DOI":"10.1016\/j.eswa.2020.113428","article-title":"An efficient Harris hawks-inspired image segmentation method","volume":"155","author":"E Rodr\u00edguez-Esparza","year":"2020","journal-title":"Expert Systems with Applications"},{"key":"pone.0315842.ref022","doi-asserted-by":"crossref","first-page":"115805","DOI":"10.1016\/j.eswa.2021.115805","article-title":"CovH2SD: A COVID-19 detection approach based on Harris Hawks Optimization and stacked deep learning","volume":"186","author":"HM Balaha","year":"2021","journal-title":"Expert systems with applications"},{"key":"pone.0315842.ref023","doi-asserted-by":"crossref","first-page":"110494","DOI":"10.1016\/j.knosys.2023.110494","article-title":"ECH3OA: an enhanced chimp-harris hawks optimization algorithm for copyright protection in Color images using watermarking techniques","volume":"269","author":"H Fahmy","year":"2023","journal-title":"Knowledge-Based Systems"},{"key":"pone.0315842.ref024","doi-asserted-by":"crossref","first-page":"114901","DOI":"10.1016\/j.eswa.2021.114901","article-title":"IWOSSA: An improved whale optimization salp swarm algorithm for solving optimization problems","volume":"176","author":"MM Saafan","year":"2021","journal-title":"Expert systems with applications"},{"issue":"18","key":"pone.0315842.ref025","doi-asserted-by":"crossref","first-page":"15907","DOI":"10.1007\/s00521-022-07241-1","article-title":"A multi-variate heart disease optimization and recognition framework","volume":"34","author":"HM Balaha","year":"2022","journal-title":"Neural Computing and Applications"},{"issue":"5","key":"pone.0315842.ref026","doi-asserted-by":"crossref","first-page":"3455","DOI":"10.1007\/s00500-019-04106-z","article-title":"Applying hybrid genetic\u2013PSO technique for tuning an adaptive PID controller used in a chemical process","volume":"24","author":"EM El-Gendy","year":"2020","journal-title":"Soft Computing"},{"key":"pone.0315842.ref027","first-page":"2","volume-title":"Encyclopedia of Optimization","author":"CW Commander","year":"2009"},{"issue":"3","key":"pone.0315842.ref028","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1145\/321958.321975","article-title":"P-complete approximation problems","volume":"23","author":"S Sahni","year":"1976","journal-title":"Journal of the ACM (JACM)"},{"issue":"6","key":"pone.0315842.ref029","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1145\/227683.227684","article-title":"Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming","volume":"42","author":"MX Goemans","year":"1995","journal-title":"Journal of the ACM (JACM)"},{"issue":"1","key":"pone.0315842.ref030","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1287\/ijoc.1080.0275","article-title":"Advanced scatter search for the max-cut problem","volume":"21","author":"R Mart\u00ed","year":"2009","journal-title":"INFORMS Journal on Computing"},{"issue":"9","key":"pone.0315842.ref031","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1016\/j.pnsc.2008.04.006","article-title":"An ant colony algorithm for solving Max-cut problem","volume":"18","author":"L Gao","year":"2008","journal-title":"Progress in Natural Science"},{"issue":"2","key":"pone.0315842.ref032","doi-asserted-by":"crossref","first-page":"298","DOI":"10.3390\/math8020298","article-title":"A deep learning algorithm for the max-cut problem based on pointer network structure with supervised learning and reinforcement learning strategies","volume":"8","author":"S Gu","year":"2020","journal-title":"Mathematics"},{"issue":"3","key":"pone.0315842.ref033","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.orl.2021.02.006","article-title":"Greedy differencing edge-contraction heuristic for the max-cut problem","volume":"49","author":"R Hassin","year":"2021","journal-title":"Operations Research Letters"},{"issue":"3","key":"pone.0315842.ref034","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1007\/s11128-024-04286-0","article-title":"Recursive QAOA outperforms the original QAOA for the MAX-CUT problem on complete graphs","volume":"23","author":"E Bae","year":"2024","journal-title":"Quantum Information Processing"},{"key":"pone.0315842.ref035","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/1090.001.0001","volume-title":"Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence","author":"JH Holland","year":"1992"},{"issue":"4598","key":"pone.0315842.ref036","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by simulated annealing","volume":"220","author":"S Kirkpatrick","year":"1983","journal-title":"science"},{"issue":"1","key":"pone.0315842.ref037","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/3477.484436","article-title":"The Any System Optimization by a colony of cooperating agents","volume":"26","author":"M Dorigo","year":"1996","journal-title":"IEEE Trans System, Man & Cybernetics-Part B"},{"key":"pone.0315842.ref038","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/j.future.2019.02.028","article-title":"Harris hawks optimization: Algorithm and applications","volume":"97","author":"AA Heidari","year":"2019","journal-title":"Future generation computer systems"},{"issue":"3","key":"pone.0315842.ref039","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/TEVC.2009.2033580","article-title":"Chemical-reaction-inspired metaheuristic for optimization","volume":"14","author":"AY Lam","year":"2009","journal-title":"IEEE transactions on evolutionary computation"},{"key":"pone.0315842.ref040","volume-title":"Nature-inspired metaheuristic algorithms","author":"XS Yang","year":"2010"},{"issue":"5","key":"pone.0315842.ref041","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/0305-0548(86)90048-1","article-title":"Future paths for integer programming and links to artificial intelligence","volume":"13","author":"F Glover","year":"1986","journal-title":"Computers & operations research"},{"key":"pone.0315842.ref042","doi-asserted-by":"crossref","unstructured":"Yang XS, Deb S. Cuckoo search via L\u00e9vy flights. In: 2009 World congress on nature & biologically inspired computing (NaBIC). Ieee; 2009. p. 210\u2013214.","DOI":"10.1109\/NABIC.2009.5393690"},{"key":"pone.0315842.ref043","unstructured":"Karaboga D, et al. An idea based on honey bee swarm for numerical optimization. Technical report-tr06, Erciyes university, engineering faculty, computer \u2026; 2005."},{"key":"pone.0315842.ref044","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","author":"S Mirjalili","year":"2014","journal-title":"Advances in engineering software"},{"key":"pone.0315842.ref045","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/978-3-642-12538-6_6","volume-title":"Nature inspired cooperative strategies for optimization (NICSO 2010)","author":"XS Yang","year":"2010"},{"key":"pone.0315842.ref046","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The whale optimization algorithm","volume":"95","author":"S Mirjalili","year":"2016","journal-title":"Advances in engineering software"},{"key":"pone.0315842.ref047","volume-title":"Nature-inspired metaheuristic algorithms","author":"XS Yang","year":"2010"},{"issue":"2","key":"pone.0315842.ref048","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1002\/j.1538-7305.1970.tb01770.x","article-title":"An efficient heuristic procedure for partitioning graphs","volume":"49","author":"BW Kernighan","year":"1970","journal-title":"The Bell system technical journal"},{"key":"pone.0315842.ref049","first-page":"1","article-title":"IHHO: an improved Harris Hawks optimization algorithm for solving engineering problems","author":"DT Akl","year":"2024","journal-title":"Neural Computing and Applications"},{"issue":"3","key":"pone.0315842.ref050","doi-asserted-by":"crossref","first-page":"1981","DOI":"10.1007\/s10586-021-03304-5","article-title":"An efficient harris hawk optimization algorithm for solving the travelling salesman problem","volume":"25","author":"FS Gharehchopogh","year":"2022","journal-title":"Cluster Computing"},{"key":"pone.0315842.ref051","doi-asserted-by":"crossref","first-page":"113211","DOI":"10.1016\/j.enconman.2020.113211","article-title":"Horizontal and vertical crossover of Harris hawk optimizer with Nelder-Mead simplex for parameter estimation of photovoltaic models","volume":"223","author":"Y Liu","year":"2020","journal-title":"Energy Conversion and Management"},{"key":"pone.0315842.ref052","first-page":"1","article-title":"A novel improved Harris Hawks optimization algorithm coupled with ELM for predicting permeability of tight carbonates","author":"N Kardani","year":"2022","journal-title":"Engineering with Computers"},{"key":"pone.0315842.ref053","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.knosys.2019.01.023","article-title":"Task scheduling in cloud computing based on hybrid moth search algorithm and differential evolution","volume":"169","author":"M Abd Elaziz","year":"2019","journal-title":"Knowledge-Based Systems"},{"key":"pone.0315842.ref054","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.asoc.2017.04.018","article-title":"Hybrid artificial bee colony algorithm with differential evolution","volume":"58","author":"SS Jadon","year":"2017","journal-title":"Applied Soft Computing"},{"key":"pone.0315842.ref055","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1016\/j.solener.2018.10.050","article-title":"Parameter extraction of solar photovoltaic models by means of a hybrid differential evolution with whale optimization algorithm","volume":"176","author":"G Xiong","year":"2018","journal-title":"Solar Energy"},{"issue":"3","key":"pone.0315842.ref056","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1137\/S1052623497328987","article-title":"A spectral bundle method for semidefinite programming","volume":"10","author":"C Helmberg","year":"2000","journal-title":"SIAM Journal on Optimization"},{"key":"pone.0315842.ref057","doi-asserted-by":"crossref","unstructured":"Xu Y, Cui Z, Wang L. Discrete Cuckoo Search with Local Search for Max-cut Problem. In: Intelligence Science I: Second IFIP TC 12 International Conference, ICIS 2017, Shanghai, China, October 25-28, 2017, Proceedings 2. Springer; 2017. p. 66\u201374.","DOI":"10.1007\/978-3-319-68121-4_7"}],"container-title":["PLOS ONE"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0315842","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T13:27:58Z","timestamp":1735565278000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0315842"}},"subtitle":[],"editor":[{"given":"Ali Safaa","family":"Sadiq","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2024,12,30]]},"references-count":57,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12,30]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0315842","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,30]]}}}