[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84685-en":3,"doc-seo-84685-105":30,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84685,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Scheduling Tasks towards Energy Autarky Benefits and Computational Costs of Flexibility","Scheduling energy-consuming jobs with time windows under a forecasted renewable supply and a limited battery is studied via the autarky problem: deciding whether all jobs can be completed without external energy. Job flexibility is modeled as the number of time steps allowed per job. The study proves NP-hardness already for flexibility two, derives polynomial-time solvable cases and fixed-parameter tractability under combined parameters, and shows W[1]-hardness for maximum flexibility alone. An ILP minimizes required external energy and experiments show that greater flexibility reduces external energy at moderate runtime costs.","arXiv :2607 .03260v 1 [ cs .DM] 3 Jul 2026  \nScheduling Tasks towards Energy Autarky: Benefits and Computational Costs of Flexibility Robert Bredereck \\#   \nInstitut für Informatik, TU Clausthal, Germany Till Fluschnik \\#   \nHumboldt-Universität zu Berlin, Department of Computer Science, Algorithm Engineering Group, Germany  \nKlaus Heeger \\#   \nDepartment of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel  \n~~ Abstract ~~  \nWe study the autarky problem: given an energy forecast, a battery, and a set of energy-consuming jobs with time windows, decide whether all jobs can be scheduled without requiring external energy. We analyze the problem through the lens of job flexibility, defined as the number of time steps at which a job may be scheduled. We show that the problem is NP-hard already for flexibility two, even in restricted settings. On the positive side, we identify settings in which the problem is polynomial-time solvable, even for large flexibilities. Moreover, we obtain fixed-parameter tractability for combined parameters involving flexibility, such as the number of jobs. In contrast, we establish W-hardness when parameterized by maximum flexibility alone, even in a restricted setting. To complement our theoretical results, we formulate an integer linear program (ILP) that computes the minimum required external energy and evaluate it experimentally on instances derived from real-world energy-consumption and radiation data. The experiments indicate that increased job flexibility substantially reduces the need for external energy at moderate computational cost.  \n2012 ACM Subject Classification Theory of computation → Design & analysis of algorithms; Mathematics of computing → Combinatorial optimization; Social & professional topics → Sustainability  \nKeywords and phrases computational sustainability, NP-hardness, parameterized complexity and algorithms, integer linear programming  \nFunding Till Fluschnik: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project “Parameterized Algorithmics in Computational Sustainability (PACS)”—FL 1247/1-1, 522475669 .  \nAcknowledgements We gratefully acknowledge Andreas Reinhardt and Mazen Bouchur (TU Clausthal, Energy Informatics group) for their continuous guidance throughout the project regarding details and specifics about energy systems.  \n2 Scheduling Tasks towards Energy Autarky  \n 1  Introduction  \nRenewable, weather-dependent resources such as solar and wind become increasingly important for energy production. This, in turn, increases the importance of reliable forecasts. Given a reliable forecast of resource availability, energy-consuming jobs (abstracting tasks or devices, e.g., a kettle or a PC) can be scheduled accordingly to directly consume available power. This applies both at a local level (e.g., solar panels on private properties) and at a global level (e.g., offshore wind farms for industrial use) . In this work, we study the following problem: given an energy forecast, a battery, and a set of energy-consuming jobs, how can the jobs be scheduled so as to minimize the required external energy—and, in particular, to decide whether no external energy is required at all. The difficulty stems from the interaction between cumulative energy constraints over time and execution-window constraints, which together create long-range dependencies between scheduling decisions.  \nOur problem is closely linked to the evaluation of a household’s degree of autarky. Several approaches to this already exist. What is novel in our approach is that we optimally solve the underlying scheduling problem arising from the fact that jobs often allow some flexibility in their execution. In particular, we show that while flexibility makes the problem computationally hard, it can yield significant energy savings in practice.  \nThe application domain ranges from households to quarters (i.e., neighborhoods) and industrial set","cbCaicFv9xTVWIvy","https://ap.wps.com/l/cbCaicFv9xTVWIvy","pdf",1736217,5,1,29,"English","en",105,"# Introduction\n## Energy autarky problem and scheduling model\n## Computational and parameterized complexity contributions\n## Integer linear program and experimental evaluation","[{\"question\":\"What is the energy autarky scheduling problem studied in the document?\",\"answer\":\"The problem determines whether all energy-consuming jobs with time windows can be scheduled using only the forecasted energy supply and a battery, without requiring any external energy.\"},{\"question\":\"How does job flexibility affect computational difficulty and energy needs?\",\"answer\":\"Flexibility can make the problem computationally hard (NP-hardness already at flexibility two), yet it can substantially reduce the minimum required external energy in experiments.\"},{\"question\":\"What algorithmic tools are provided beyond complexity results?\",\"answer\":\"An integer linear program (ILP) is formulated to compute the minimum required external energy, and the document evaluates it experimentally on instances derived from real-world energy-consumption and radiation 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is the energy autarky scheduling problem studied in the document?","Question",{"text":76,"@type":77},"The problem determines whether all energy-consuming jobs with time windows can be scheduled using only the forecasted energy supply and a battery, without requiring any external energy.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does job flexibility affect computational difficulty and energy needs?",{"text":81,"@type":77},"Flexibility can make the problem computationally hard (NP-hardness already at flexibility two), yet it can substantially reduce the minimum required external energy in experiments.",{"name":83,"@type":74,"acceptedAnswer":84},"What algorithmic tools are provided beyond complexity results?",{"text":85,"@type":77},"An integer linear program (ILP) is formulated to compute the minimum required external energy, and the document evaluates it experimentally on instances derived from real-world energy-consumption and radiation 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