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Hardness of Approximation Between P and NP (Paperback)
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Hardness of Approximation Between P and NP (Paperback)
Series: ACM Books
Expected to ship within 10 - 15 working days
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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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