[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84854-en":3,"doc-seo-84854-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84854,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Data-dependent Evaluations for Budgeted Submodular Maximization","Submodular maximization supports optimization approaches across machine learning and data mining, yet its NP-hardness makes classical analysis rely on worst-case approximation factors that do not reflect instance quality. This paper develops new data-dependent upper bounds for budgeted submodular maximization under a knapsack constraint. Theoretical results show the proposed bounds dominate the optimal value, while experiments on real-world datasets demonstrate improved ability to certify solution-to-optimum closeness for specific instances.","arXiv :2607 .05759v 1 [ cs .DS] 7 Jul 2026  \nData-dependent Evaluations for Budgeted Submodular  \nMaximization  \nLejian Zhang 1 , Xueyan Tang 1 , and Jing Tang2  \n1 College of Computing and Data Science, Nanyang Technological University, Singapore  \n2 Data Science and Analytics Thrust, The Hong Kong University of Science  \nand Technology (Guangzhou), China  \n[lejian001@e.ntu.edu.sg](lejian001@e.ntu.edu.sg), [asxytang@ntu.edu.sg](asxytang@ntu.edu.sg), [jingtang@ust.hk](jingtang@ust.hk)  \nAbstract  \nSubmodular maximization is an important building block for developing algorithms in many areas such as machine learning and data mining. Due to the NP-hardness of the problem, analysis of submodular maximization algorithms typically provides pessimistic worst-case approximation factors only. It is not easy to evaluate how close a produced solution is to an optimal one for a given problem instance. In this paper, we develop new data-dependent upper bounds for submodular maximization with a knapsack constraint. We theoretically prove that they dominate the optimal solution and empirically demonstrate their advantages in certifying how close to optimal a solution is through experiments with real-world datasets.  \n1 Introduction  \nA great number of optimization problems in different areas can be modeled as submodular maximization problems, including facility location [21], feature selection [8], recommendation system [18], influence maximization [26], document summarization [20], maximum coverage [10], sensor placement [16], exemplar sampling [12] and market expansion [9] .  \nDue to the NP-hard nature of submodular maximization, many approximation algorithms have been developed, attempting to find good solutions in polynomial time. Approximation algorithms are typically evaluated by the approximation factor, which refers to the worst-case ratio between the function value of an output solution and that of an optimal solution. Nemhauser and Wolsey [22] introduced a simple greedy algorithm which guarantees an approximation factor of 1 − 1/e for the problem of Monotone Submodular Maximization with a Cardinality constraint (MSMC) (finding a set S of a given size which maximizes f(S)) . But the greedy algorithm does not guarantee any positive approximation factor for the more general problem of Monotone Submodular Maximization with a Knapsack constraint (MSMK)(finding a set S of total cost at most a given budget which maximizes f(S), assuming elements have costs) . Wolsey [27] improved the simple greedy algorithm by a small modification and showed that it achieves an approximation factor of 0.357 . Khuller et al. [15] attempted to derive an approximation factor of 1 − 1/ √e ≈ 0.393 for the modified greedy algorithm, but their analysis was found flawed [29] . Tang et al. [25] proved that this modified greedy algorithm actually guarantees an approximation factor of 0.405 . Subsequently, more careful approximation analysis showed that the exact approximation factor of the modified greedy algorithm is between 0.427 and 0.42945 [11, 17] . Both the simple and modified greedy algorithms run in O (n2 ) time, where n is the size of the ground set. Sviridenko [24] proposed  \na (1 − 1/e)-approximation algorithm at the expense of increasing the time complexity to O (n5 ) . The time complexity of the algorithm was later reduced to O (n4 ) without sacrificing the approximation factor [11, 17] . Yaroslavtsev et al. [28] developed a 0.5-approximation algorithm called Greedy+Max that runs in O (n2 ) time. Feldman et al. [11] augmented Greedy+Max with a single guess to achieve an approximation factor of˜0.6174 in O (n3 ) running time. The running times of these algorithms can  \nbe reduced from O (ni) to O(ni−1/ϵ) (ignoring poly-logarithmic terms) at the cost of losing an ϵ in the approximation factors using the threshold technique of [2] .  \nAlthough the above algorithms provide approximation factors, these constant factors usually leave a significant gap betwe","cbCaig3HvPykQjlS","https://ap.wps.com/l/cbCaig3HvPykQjlS","pdf",669344,1,26,"English","en",105,"# Introduction\n## Problem background and prior approximation results\n## Data-dependent bounds and motivation\n## Contributions and paper organization","[{\"question\":\"Why are constant worst-case approximation factors insufficient for practical evaluation?\",\"answer\":\"They measure only worst-case performance and can leave a large gap between the obtained solution quality and the optimal value for a specific instance.\"},{\"question\":\"What new contributions does the paper make for budgeted submodular maximization with a knapsack constraint?\",\"answer\":\"It proposes two strategies—slicing and removing—to build tighter data-dependent upper bounds that dominate the optimal solution, and it converts them into linear programs to incorporate multiple base sets.\"},{\"question\":\"How are the proposed data-dependent bounds validated in the paper?\",\"answer\":\"The paper runs extensive experiments on real-world applications such as maximum coverage, revenue maximization, and feature selection, demonstrating improved empirical certification of closeness to optimality.\"}]",1784198831,66,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"data-dependent-evaluations-for-budgeted-submodular-maximization","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/data-dependent-evaluations-for-budgeted-submodular-maximization/84854/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why are constant worst-case approximation factors insufficient for practical evaluation?","Question",{"text":75,"@type":76},"They measure only worst-case performance and can leave a large gap between the obtained solution quality and the optimal value for a specific instance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What new contributions does the paper make for budgeted submodular maximization with a knapsack constraint?",{"text":80,"@type":76},"It proposes two strategies—slicing and removing—to build tighter data-dependent upper bounds that dominate the optimal solution, and it converts them into linear programs to incorporate multiple base sets.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the proposed data-dependent bounds validated in the paper?",{"text":84,"@type":76},"The paper runs extensive experiments on real-world applications such as maximum coverage, revenue maximization, and feature selection, demonstrating improved empirical certification of closeness to 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