## Hardness of approximation between P and NP [electronic resources] / Aviad Rubinstein.

Material type: TextSeries: ACM books ; #24Publication details: [New York] : Association for Computing Machinery, c2019Description: 1 online resources (xv, 301 pages) : illustrations (chiefly color)ISBN: 9781947487222 (epub); 9781947487215 (pdf)Subject(s): NP-complete problems | Equilibrium | Computer algorithms | Programming (Mathematics) | Game theory | Computational complexity | Mathematical optimizationDDC classification: 511.352 LOC classification: QA267.7 | .R83 2019ebOnline resources: Available in ACM Digital Library. Requires Log In to view full text.Item type | Current library | Collection | Call number | Copy number | Status | Date due | Barcode |
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General Circulation | APU Library Online Database | E-Book | QA267.7 .R83 2019eb (Browse shelf (Opens below)) | 1 | Available |

Revision of author's thesis (doctoral)--University of California, Berkeley, 2017.

Includes bibliographical references and index.

part 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 equilibrium

2. 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 lemmata

part 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 games

4. Brouwer's fixed point -- 4.1. Brouwer with �[infinity] -- 4.2. Euclidean Brouwer

part III. PPAD -- 5. PPAD-hardness of approximation

6. The generalized circuit problem -- 6.1. Proof overview -- 6.2. From Brouwer to [epsilon]-Gcircuit -- 6.3. Gcircuit with fan-out 2

7. Many-player games -- 7.1. Graphical, polymatrix games -- 7.2. Succinct games

8. 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 inapproximability

10. CourseMatch -- 10.1. The course allocation problem -- 10.2. A-CEEI is PPAD-hard -- 10.3. A-CEEI [epsilon] PPAD

part IV. Quasi-polynomial time -- 11. Birthday repetition -- 11.1. Warm-up : best [epsilon]-Nash

12. Densest k-subgraph -- 12.1. Construction (and completeness) -- 12.2. Soundness

13. Community detection -- 13.1. Related works -- 13.2. Overview of proofs -- 13.3. Hardness of counting communities -- 13.4. Hardness of detecting communities

14. 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 dimension

15. Signaling -- 15.1. Techniques -- 15.2. Near-optimal signaling is hard

part 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.

Nash 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.

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