06589nam a22005897a 4500003000400000005001700004008004100021015001900062016001800081020002500099020002400124040002100148042001100169050002500180082001600205100003000221245009300251260006400344300007500408490003400483500008600517504005100603505021000654505022400864505028901088505009001377505005601467505013701523505008401660505025901744505011902003505010102122505008702223505016402310505026702474505007802741505025402819520235703073538003605430538004705466650003305513650002405546650002505570650003505595650001705630650003005647650003105677856013705708942001605845999001905861952011905880APU20221101135833.0210804s2019 enka b 001 0 eng d aGBB9F58342bnb7 a0195358842Uk a9781947487222 (epub) a9781947487215 (pdf) aUkbengcAPUdSF aukblsr aQA267.7b.R83 2019eb04a511.3522231 aRubinstein, Aviad,94744010aHardness of approximation between P and NP h[electronic resources] /cAviad Rubinstein. a[New York] : bAssociation for Computing Machinery,cc2019. a1 online resources (xv, 301 pages) :billustrations (chiefly color). ;0 aACM books ;v#24x2374-6777 ; aRevision of author's thesis (doctoral)--University of California, Berkeley, 2017. aIncludes bibliographical references and index.0 apart I. Overview -- 1. The frontier of intractability -- 1.1. PPAD : finding a needle you know is in the haystack -- 1.2. Quasi-polynomial time and the birthday paradox -- 1.3. Approximate Nash equilibrium8 a2. Preliminaries -- 2.1. Nash equilibrium and relaxations -- 2.2. PPAD and end-of-a-line -- 2.3. Exponential time hypotheses -- 2.4. PCP theorems -- 2.5. Learning theory -- 2.6. Information theory -- 2.7. Useful lemmata8 apart II. Communication complexity -- 3. Communication complexity of approximate Nash equilibrium -- 3.1. Uncoupled dynamics -- 3.2. Techniques -- 3.3. Additional related literature -- 3.4. Proof overview -- 3.5. Proofs -- 3.6. An open problem : correlated equilibria in 2-player games8 a4. Brouwer's fixed point -- 4.1. Brouwer with �[infinity] -- 4.2. Euclidean Brouwer8 apart III. PPAD -- 5. PPAD-hardness of approximation8 a6. The generalized circuit problem -- 6.1. Proof overview -- 6.2. From Brouwer to [epsilon]-Gcircuit -- 6.3. Gcircuit with fan-out 28 a7. Many-player games -- 7.1. Graphical, polymatrix games -- 7.2. Succinct games8 a8. Bayesian Nash equilibrium -- 9. Market equilibrium -- 9.1. Why are non-monotone markets hard? -- 9.2. High-level structure of the proof -- 9.3. Adaptations for constant factor inapproximability -- 9.4. Non-monotone markets : proof of inapproximability8 a10. CourseMatch -- 10.1. The course allocation problem -- 10.2. A-CEEI is PPAD-hard -- 10.3. A-CEEI [epsilon] PPAD8 apart IV. Quasi-polynomial time -- 11. Birthday repetition -- 11.1. Warm-up : best [epsilon]-Nash8 a12. Densest k-subgraph -- 12.1. Construction (and completeness) -- 12.2. Soundness8 a13. Community detection -- 13.1. Related works -- 13.2. Overview of proofs -- 13.3. Hardness of counting communities -- 13.4. Hardness of detecting communities8 a14. VC and Littlestone's dimensions -- 14.1. Discussion -- 14.2. Techniques -- 14.3. Related Work -- 14.4. Inapproximability of the VC dimension -- 14.5. Inapproximability of Littlestone's dimension -- 14.6. Quasi-polynomial algorithm for Littlestone's dimension8 a15. Signaling -- 15.1. Techniques -- 15.2. Near-optimal signaling is hard8 apart V. Approximate Nash equilibrium -- 16. 2-player approximate Nash equilibrium -- 16.1. Technical overview -- 16.2. End-of-a-line with local computation -- 16.3. Holographic proof -- 16.4. Polymatrix WeakNash -- 16.5. From Polymatrix to Bimatrix. aNash equilibrium is the central solution concept in Game Theory. Since Nash's original paper in 1951, it has found countless applications in modeling strategic behavior of traders in markets, (human) drivers and (electronic) routers in congested networks, nations in nuclear disarmament negotiations, and more. A decade ago, the relevance of this solution concept was called into question by computer scientists, who proved (under appropriate complexity assumptions) that computing a Nash equilibrium is an intractable problem. And if centralized, specially designed algorithms cannot find Nash equilibria, why should we expect distributed, selfish agents to converge to one? The remaining hope was that at least approximate Nash equilibria can be efficiently computed. Understanding whether there is an efficient algorithm for approximate Nash equilibrium has been the central open problem in this field for the past decade. In this book, we provide strong evidence that even finding an approximate Nash equilibrium is intractable. We prove several intractability theorems for different settings (two-player games and many-player games) and models (computational complexity, query complexity, and communication complexity). In particular, our main result is that under a plausible and natural complexity assumption ("Exponential Time Hypothesis for PPAD"), there is no polynomial-time algorithm for finding an approximate Nash equilibrium in two-player games. The problem of approximate Nash equilibrium in a two-player game poses a unique technical challenge: it is a member of the class PPAD, which captures the complexity of several fundamental total problems, i.e., problems that always have a solution; and it also admits a quasipolynomial time algorithm. Either property alone is believed to place this problem far below NP-hard problems in the complexity hierarchy; having both simultaneously places it just above P, at what can be called the frontier of intractability. Indeed, the tools we develop in this book to advance on this frontier are useful for proving hardness of approximation of several other important problems whose complexity lies between P and NP: Brouwer's fixed point, market equilibrium, CourseMatch (A-CEEI), densest k-subgraph, community detection, VC dimension and Littlestone dimension, and signaling in zero-sum games. aMode of access: World Wide Web. aSystem requirements: Adobe Acrobat Reader. 0aNP-complete problems.947441 0aEquilibrium.947442 0aComputer algorithms. 0aProgramming (Mathematics)9488 0aGame theory. 0aComputational complexity. 0aMathematical optimization. uhttps://dl-acm-org.ezproxy.apu.edu.my/doi/book/10.1145/3241304zAvailable in ACM Digital Library. Requires Log In to view full text. 2lcccE-Book c383497d383497 00102lcc40708E-BookaAPUbAPUcOND d2022-03-08eOTHERSl0oQA267.7 .R83 2019ebr2022-03-08t1w2022-03-08yGC