Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos
{"title":"纯电路:PPAD 的严格不可逼近性","authors":"Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos","doi":"10.1145/3678166","DOIUrl":null,"url":null,"abstract":"\n The current state-of-the-art methods for showing inapproximability in\n PPAD\n arise from the ε-Generalized-Circuit (ε-\n GCircuit\n ) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-\n GCircuit\n is\n PPAD\n -hard, and subsequent work has shown hardness results for other problems in\n PPAD\n by using ε-\n GCircuit\n as an intermediate problem.\n \n \n We introduce\n Pure-Circuit\n , a new intermediate problem for\n PPAD\n , which can be thought of as ε-\n GCircuit\n pushed to the limit as ε → 1, and we show that the problem is\n PPAD\n -complete. We then prove that ε-\n GCircuit\n is\n PPAD\n -hard for all ε < 1/10 by a reduction from\n Pure-Circuit\n , and thus strengthen all prior work that has used\n GCircuit\n as an intermediate problem from the existential-constant regime to the large-constant regime.\n \n \n We show that stronger inapproximability results can be derived by reducing directly from\n Pure-Circuit\n . In particular, we prove tight inapproximability results for computing approximate Nash equilibria and approximate well-supported Nash equilibria in graphical games, for finding approximate well-supported Nash equilibria in polymatrix games, and for finding approximate equilibria in threshold games.\n","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.3000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Pure-Circuit: Tight Inapproximability for PPAD\",\"authors\":\"Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos\",\"doi\":\"10.1145/3678166\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n The current state-of-the-art methods for showing inapproximability in\\n PPAD\\n arise from the ε-Generalized-Circuit (ε-\\n GCircuit\\n ) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-\\n GCircuit\\n is\\n PPAD\\n -hard, and subsequent work has shown hardness results for other problems in\\n PPAD\\n by using ε-\\n GCircuit\\n as an intermediate problem.\\n \\n \\n We introduce\\n Pure-Circuit\\n , a new intermediate problem for\\n PPAD\\n , which can be thought of as ε-\\n GCircuit\\n pushed to the limit as ε → 1, and we show that the problem is\\n PPAD\\n -complete. We then prove that ε-\\n GCircuit\\n is\\n PPAD\\n -hard for all ε < 1/10 by a reduction from\\n Pure-Circuit\\n , and thus strengthen all prior work that has used\\n GCircuit\\n as an intermediate problem from the existential-constant regime to the large-constant regime.\\n \\n \\n We show that stronger inapproximability results can be derived by reducing directly from\\n Pure-Circuit\\n . In particular, we prove tight inapproximability results for computing approximate Nash equilibria and approximate well-supported Nash equilibria in graphical games, for finding approximate well-supported Nash equilibria in polymatrix games, and for finding approximate equilibria in threshold games.\\n\",\"PeriodicalId\":50022,\"journal\":{\"name\":\"Journal of the ACM\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the ACM\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1145/3678166\",\"RegionNum\":2,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, HARDWARE & ARCHITECTURE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the ACM","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3678166","RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, HARDWARE & ARCHITECTURE","Score":null,"Total":0}
The current state-of-the-art methods for showing inapproximability in
PPAD
arise from the ε-Generalized-Circuit (ε-
GCircuit
) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-
GCircuit
is
PPAD
-hard, and subsequent work has shown hardness results for other problems in
PPAD
by using ε-
GCircuit
as an intermediate problem.
We introduce
Pure-Circuit
, a new intermediate problem for
PPAD
, which can be thought of as ε-
GCircuit
pushed to the limit as ε → 1, and we show that the problem is
PPAD
-complete. We then prove that ε-
GCircuit
is
PPAD
-hard for all ε < 1/10 by a reduction from
Pure-Circuit
, and thus strengthen all prior work that has used
GCircuit
as an intermediate problem from the existential-constant regime to the large-constant regime.
We show that stronger inapproximability results can be derived by reducing directly from
Pure-Circuit
. In particular, we prove tight inapproximability results for computing approximate Nash equilibria and approximate well-supported Nash equilibria in graphical games, for finding approximate well-supported Nash equilibria in polymatrix games, and for finding approximate equilibria in threshold games.
期刊介绍:
The best indicator of the scope of the journal is provided by the areas covered by its Editorial Board. These areas change from time to time, as the field evolves. The following areas are currently covered by a member of the Editorial Board: Algorithms and Combinatorial Optimization; Algorithms and Data Structures; Algorithms, Combinatorial Optimization, and Games; Artificial Intelligence; Complexity Theory; Computational Biology; Computational Geometry; Computer Graphics and Computer Vision; Computer-Aided Verification; Cryptography and Security; Cyber-Physical, Embedded, and Real-Time Systems; Database Systems and Theory; Distributed Computing; Economics and Computation; Information Theory; Logic and Computation; Logic, Algorithms, and Complexity; Machine Learning and Computational Learning Theory; Networking; Parallel Computing and Architecture; Programming Languages; Quantum Computing; Randomized Algorithms and Probabilistic Analysis of Algorithms; Scientific Computing and High Performance Computing; Software Engineering; Web Algorithms and Data Mining