{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T07:47:05Z","timestamp":1757576825839,"version":"3.37.3"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971146"],"award-info":[{"award-number":["11971146"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003787","name":"Natural Science Foundation of Hebei Province","doi-asserted-by":"publisher","award":["A2023205009"],"award-info":[{"award-number":["A2023205009"]}],"id":[{"id":"10.13039\/501100003787","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Interdisciplinary Research Foundation of Hebei Normal University","award":["L2024J01"],"award-info":[{"award-number":["L2024J01"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Comb Optim"],"published-print":{"date-parts":[[2024,10]]},"DOI":"10.1007\/s10878-024-01212-z","type":"journal-article","created":{"date-parts":[[2024,10,11]],"date-time":"2024-10-11T18:01:26Z","timestamp":1728669686000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["W-prize-collecting scheduling problem on parallel machines"],"prefix":"10.1007","volume":"48","author":[{"given":"Bo","family":"Hou","sequence":"first","affiliation":[]},{"given":"Tianjiao","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Suogang","family":"Gao","sequence":"additional","affiliation":[]},{"given":"Guanghua","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Weili","family":"Wu","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6872-3315","authenticated-orcid":false,"given":"Wen","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,11]]},"reference":[{"key":"1212_CR1","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1137\/S0895480196300522","volume":"13","author":"Y Bartal","year":"2000","unstructured":"Bartal Y, Leonardi S, Marchetti-Spaccamela A, Sgall JSL (2000) Multiprocessor scheduling with rejection. SIAM J Discret Math 13:64\u201378","journal-title":"SIAM J Discret Math"},{"issue":"9","key":"1212_CR2","doi-asserted-by":"publisher","first-page":"1563","DOI":"10.1002\/j.1538-7305.1966.tb01709.x","volume":"45","author":"RL Graham","year":"1966","unstructured":"Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell Syst Tech J 45(9):1563\u20131581","journal-title":"Bell Syst Tech J"},{"key":"1212_CR3","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1016\/S0167-5060(08)70356-X","volume":"5","author":"RL Graham","year":"1979","unstructured":"Graham RL, Lawler EL, Lenstra JK, Kan AR (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discr Math 5:287\u2013326","journal-title":"Ann Discr Math"},{"key":"1212_CR4","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s11590-017-1135-8","volume":"13","author":"L Han","year":"2017","unstructured":"Han L, Xu D, Du D, Wu C (2017) A 5-approximation algorithm for the $$k$$-prize-collecting Steiner tree problem. Optim Lett 13:573\u2013585","journal-title":"Optim Lett"},{"key":"1212_CR5","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.tcs.2020.07.014","volume":"844","author":"X Hou","year":"2020","unstructured":"Hou X, Liu W, Hou B (2020) An approximation algorithm for the $$k$$-prize-collecting multicut on a tree problem. Theor Comput Sci 844:26\u201333","journal-title":"Theor Comput Sci"},{"key":"1212_CR6","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1016\/j.tcs.2015.10.007","volume":"607","author":"W Li","year":"2015","unstructured":"Li W, Li J, Zhang X, Chen Z (2015) Penalty cost constrained identical parallel machine scheduling problem. Theor Comput Sci 607:181\u2013192","journal-title":"Theor Comput Sci"},{"key":"1212_CR7","doi-asserted-by":"publisher","first-page":"2165","DOI":"10.1007\/s11590-021-01724-1","volume":"15","author":"X Liu","year":"2021","unstructured":"Liu X, Li W (2021) Approximation algorithms for the multiprocessor scheduling with submodular penalties. Optim Lett 15:2165\u20132180","journal-title":"Optim Lett"},{"key":"1212_CR8","doi-asserted-by":"publisher","first-page":"2373","DOI":"10.1007\/s11590-021-01831-z","volume":"16","author":"X Liu","year":"2022","unstructured":"Liu X, Li W, Xie R (2022) A prime-dual approximation algorithm for the $$k$$-prize-collecting minimum power cover problem. Optim Lett 16:2373\u20132385","journal-title":"Optim Lett"},{"issue":"3","key":"1212_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/s11704-022-1665-9","volume":"17","author":"X Liu","year":"2023","unstructured":"Liu X, Li W, Yang J (2023) A prime-dual approximation algorithm for the $$k$$-prize-collecting minimum vertex cover problem with submodular penalties. Front Comput Sci 17(3):173404","journal-title":"Front Comput Sci"},{"key":"1212_CR10","doi-asserted-by":"crossref","unstructured":"Sun R, Deng B (2022) $$W$$-prize-collecting scheduling problem on a single machine. In: proceedings of the 2022 6th international conference on computer science and artificial intelligence, pp. 237\u2013242","DOI":"10.1145\/3577530.3577568"},{"key":"1212_CR11","doi-asserted-by":"crossref","unstructured":"Wang W, Liu X (2022) A combinatorial 2-approximation algorithm for the parallel-machine scheduling with release times and submodular penalties. Mathematics 10:61","DOI":"10.3390\/math10010061"},{"key":"1212_CR12","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s10288-016-0304-4","volume":"14","author":"L Zhang","year":"2016","unstructured":"Zhang L, Lu L (2016) Parallel-machine scheduling with release dates and rejection. 4OR 14:165\u2013172","journal-title":"4OR"},{"key":"1212_CR13","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s10878-021-00842-x","volume":"44","author":"H Zheng","year":"2022","unstructured":"Zheng H, Gao S, Liu W, Wu W, Du D, Hou B (2022) Approximation algorithm for the parallel-machine scheduling problem with release dates and submodular rejection penalties. J Comb Optim 44:343\u2013353","journal-title":"J Comb Optim"}],"container-title":["Journal of Combinatorial Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-024-01212-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10878-024-01212-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-024-01212-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,17]],"date-time":"2024-10-17T19:10:34Z","timestamp":1729192234000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10878-024-01212-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["1212"],"URL":"https:\/\/doi.org\/10.1007\/s10878-024-01212-z","relation":{},"ISSN":["1382-6905","1573-2886"],"issn-type":[{"type":"print","value":"1382-6905"},{"type":"electronic","value":"1573-2886"}],"subject":[],"published":{"date-parts":[[2024,10]]},"assertion":[{"value":"22 September 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no Conflict of interest regarding the publication of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"26"}}