Expected complexity of graph partitioning problems
Discrete Applied Mathematics - Special issue: Combinatorial Optimization 1992 (CO92)
On limited nondeterminism and the complexity of the V-C dimension
Journal of Computer and System Sciences
Finding a large hidden clique in a random graph
proceedings of the eighth international conference on Random structures and algorithms
Finding and certifying a large hidden clique in a semirandom graph
Random Structures & Algorithms
Hiding Cliques for Cryptographic Security
Designs, Codes and Cryptography
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Probable Value of the Lovász-Schrijver Relaxations for Maximum Independent Set
SIAM Journal on Computing
Graphical Models for Game Theory
UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
Playing large games using simple strategies
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Spectral Partitioning of Random Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
The complexity of computing a Nash equilibrium
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Computing Nash Equilibria: Approximation and Smoothed Complexity
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Testing k-wise and almost k-wise independence
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Computing good nash equilibria in graphical games
Proceedings of the 8th ACM conference on Electronic commerce
Approximating nash equilibria using small-support strategies
Proceedings of the 8th ACM conference on Electronic commerce
Progress in approximate nash equilibria
Proceedings of the 8th ACM conference on Electronic commerce
Complexity results about Nash equilibria
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
New algorithms for approximate Nash equilibria in bimatrix games
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
An optimization approach for approximate Nash equilibria
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Polynomial algorithms for approximating nash equilibria of bimatrix games
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
A note on approximate nash equilibria
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
HAPLOFREQ: estimating haplotype frequencies efficiently
RECOMB'05 Proceedings of the 9th Annual international conference on Research in Computational Molecular Biology
Small Clique Detection and Approximate Nash Equilibria
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
A direct reduction from k-player to 2-player approximate nash equilibrium
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
How Hard Is It to Approximate the Best Nash Equilibrium?
SIAM Journal on Computing
On the complexity of approximating a Nash equilibrium
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On the Complexity of Approximating a Nash Equilibrium
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
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The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPAD-completeness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NP-hard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NP-hardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasi-polynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a two-player game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a notoriously difficult problem in modern Combinatorics, of finding a planted (but hidden) clique in a random graph G(n, 1/2). Our reduction starts from an instance with planted clique size k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).