[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81612-en":3,"doc-seo-81612-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},81612,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","An Approximation Notion between P and FPTAS","The document introduces an approximation notion for NP-hard optimization problems based on amortized relaxation: the relaxed optimum is the largest per-copy value achievable when many independent copies of the input are solved together. The authors show that, assuming P≠NP, this new notion is strictly stronger than FPTAS while still strictly weaker than having a polynomial-time exact algorithm. The work defines a corresponding complexity class and motivates practical relevance via fractional partitioning scenarios.","arXiv :2603 . 17489v4 [ cs .CC] 10 Jul 2026  \nAn Approximation Notion between P and FPTAS  \nSamuel Bismuth \\# 􀀚  \nDepartment of Computer Science, Ariel University, Ariel 40700, Israel Erel Segal-Halevi \\# 􀀚  \nDepartment of Computer Science, Ariel University, Ariel 40700, Israel  \n~~ Abstract ~~  \nWe present an approximation notion for NP-hard optimization problems. The notion is based on an amortized relaxation: the relaxed optimum of an input is the largest per-copy value attainable when many copies of the input are solved together. We prove that (assuming P NP) the new notion is strictly stronger than FPTAS, but strictly weaker than having a polynomial-time algorithm. Our results introduce a new computational complexity class, which is a strict superset of P and a strict subset of FPTAS.  \n2012 ACM Subject Classification Mathematics of computing → Combinatorial algorithms  \nKeywords and phrases FPTAS, algorithm, complexity, combinatorial problems, approximation  \nFunding Samuel Bismuth: Israel Science Foundation grant no. 712/20 . Erel Segal-Halevi: Israel Science Foundation grant no. 712/20, 1092/24 .  \n2 An Approximation Notion between P and FPTAS  \n 1  Introduction  \nWhen an optimization problem is found to be NP-hard, we assume that it cannot be solved exactly by a polynomial-time algorithm, and look for polynomial-time approximation algorithms. The most efficient kind of an approximation algorithm currently known is the FPTAS (Fully Polynomial Time Approximation Scheme): for any ϵ > 0, it finds a solution that is at least (1 − ϵ) times the optimal solution (in case of a maximization problem), and runs in time polynomial in the input size and 1/ϵ . Schuurman and Woeginger write in their Approximation Schemes Tutorial [15] that  \n“With respect to worst case approximation, an FPTAS is the strongest possible result that we can derive for an NP-hard problem”.  \nIn this paper we challenge this claim. We show a new (to the best of our knowledge) kind of approximation algorithm, that is stronger than FPTAS in a precise sense. We call it FFPTAS—Fractional Fully Polynomial Time Approximation Scheme.  \nInstead of approximating the optimal solution value, an FFPTAS approximates the amortized optimum of the problem: the largest per-copy value that can be attained when many independent copies of the input are solved together as a single instance. The amortized optimum is a relaxation of the optimum—it is always at least as large  \n—and it plays the role that the optimum of a fractional relaxation plays in classical approximation algorithms. For any t > 0, it finds a solution with value at least (1 − t) times the amortized optimum (or asserts that such a solution does not exist), and runs in time polynomial in the input size and 1/t.  \nWe prove that (under certain conditions) FFPTAS is a strictly better approximation than FPTAS, that is: every problem that has an FFPTAS has an FPTAS, but the opposite is not true unless P=NP. We complement this result by showing an FFPTAS for an NP-hard problem. Together, our results assert the existence of a new complexity class for optimization problems, that lies strictly between P and FPTAS.  \n1.1 Motivation  \nOur main motivation for studying FFPTAS is the theoretical discovery of a new complexity class, refining the complexity hierarchy of NP-hard optimization problems.  \nBut FFPTAS might also have practical applications in situations in which a perfect solution is required, but can be attained only when allowing fractions. As an example, consider the problem of dividing items of different values between two partners. It may be required by law to give each partner exactly 1/2 of the total value. However, such a perfect partition may be impossible to attain if the items cannot be split. A possible solution is to have the partner who received the higher value compensate the other partner by monetary payments; but the amount of monetary payments available might also be bounded. For this problem, the ","cbCaip0QopKLWqzA","https://ap.wps.com/l/cbCaip0QopKLWqzA","pdf",721485,1,23,"English","en",105,"# Introduction\n## Motivation\n# Definitions","[{\"question\":\"What is the main idea behind the new approximation notion introduced in the paper?\",\"answer\":\"It approximates the amortized optimum, defined as the largest per-copy value achievable when solving many independent copies together, rather than approximating the discrete optimum value directly.\"},{\"question\":\"How does the paper compare the new notion with FPTAS and polynomial-time exact algorithms?\",\"answer\":\"Under appropriate assumptions, the new notion is strictly stronger than FPTAS (every FFPTAS problem has an FPTAS), but it is strictly weaker than having a polynomial-time exact algorithm, unless P=NP.\"},{\"question\":\"Why might an FFPTAS be useful in practical fractional scenarios?\",\"answer\":\"In settings like partitioning items where exact solutions require fractions, the amortized optimum can match the perfect fractional value, enabling near-perfect balancing through limited payments or limited fractional 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is the main idea behind the new approximation notion introduced in the paper?","Question",{"text":74,"@type":75},"It approximates the amortized optimum, defined as the largest per-copy value achievable when solving many independent copies together, rather than approximating the discrete optimum value directly.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the paper compare the new notion with FPTAS and polynomial-time exact algorithms?",{"text":79,"@type":75},"Under appropriate assumptions, the new notion is strictly stronger than FPTAS (every FFPTAS problem has an FPTAS), but it is strictly weaker than having a polynomial-time exact algorithm, unless P=NP.",{"name":81,"@type":72,"acceptedAnswer":82},"Why might an FFPTAS be useful in practical fractional scenarios?",{"text":83,"@type":75},"In settings like partitioning items where exact solutions require fractions, the amortized optimum can match the perfect fractional value, enabling near-perfect balancing through 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