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Computing Nash equilibria: Approximation and smoothed complexity
 In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2006
"... By proving that the problem of computing a 1/n Θ(1)approximate Nash equilibrium remains PPADcomplete, we show that the BIMATRIX game is not likely to have a fully polynomialtime approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫapproximate Nash ..."
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Cited by 68 (10 self)
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By proving that the problem of computing a 1/n Θ(1)approximate Nash equilibrium remains PPADcomplete, we show that the BIMATRIX game is not likely to have a fully polynomialtime approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫapproximate Nash equilibrium of an n×n bimatrix game, unless PPAD ⊆ P. Instrumental to our proof, we introduce a new discrete fixedpoint problem on a highdimensional cube with a constant sidelength, such as on an ndimensional cube with sidelength 7, and show that they are PPADcomplete. Furthermore, we prove that it is unlikely, unless PPAD ⊆ RP, that the smoothed complexity of the LemkeHowson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/σ under perturbations with magnitude σ. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.
Settling the Complexity of Computing TwoPlayer Nash Equilibria
"... We prove that Bimatrix, the problem of finding a Nash equilibrium in a twoplayer game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the c ..."
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Cited by 44 (3 self)
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We prove that Bimatrix, the problem of finding a Nash equilibrium in a twoplayer game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the complexity of fourplayer Nash equilibria [21], settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of twoplayer Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomialtime approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic LemkeHowson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: • ArrowDebreu market equilibria are PPADhard to compute.
Quantum and classical query complexities of local search are polynomially related
 In Proc. of 36th STOC
, 2004
"... Let f be an integer valued function on a finite set V. We call an undirected graph G(V, E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V, E) find a vertex x ∈ V such that f(x) is not bigger than any value that f ..."
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Cited by 16 (1 self)
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Let f be an integer valued function on a finite set V. We call an undirected graph G(V, E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V, E) find a vertex x ∈ V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x? ” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous [4] and Aaronson [1] and solves the main open problem in [1]. 1
The npcompleteness column: Finding needles in haystacks
 ACM Transactions on Algorithms
, 2007
"... Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, h ..."
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Cited by 7 (0 self)
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Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at
The Complexity of the Homotopy Method, Equilibrium Selection, and LemkeHowson Solutions, Arxiv technical report 1006.5352
, 2010
"... Abstract — We show that the widely used homotopy method for solving fixpoint problems, as well as the HarsanyiSelten equilibrium selection process for games, are PSPACEcomplete to implement. Extending our result for the HarsanyiSelten process, we show that several other homotopybased algorithms f ..."
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Cited by 2 (1 self)
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Abstract — We show that the widely used homotopy method for solving fixpoint problems, as well as the HarsanyiSelten equilibrium selection process for games, are PSPACEcomplete to implement. Extending our result for the HarsanyiSelten process, we show that several other homotopybased algorithms for finding equilibria of games are also PSPACEcomplete to implement. A further application of our techniques yields the result that it is PSPACEcomplete to compute any of the equilibria that could be found via the classical LemkeHowson algorithm, a complexitytheoretic strengthening of the result in [24]. These results show that our techniques can be widely applied and suggest that the PSPACEcompleteness of implementing homotopy methods is a general principle. Keywordsgame theory, computational complexity
COMPUTING AND PROVING WITH PIVOTS
, 2013
"... Abstract. A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solvin ..."
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Abstract. A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going from one subset σ to a new one σ ′ by deleting an element inside σ and adding an element outside σ: σ ′ = σ \ {v} ∪ {u}, with v ∈ σ and u / ∈ σ. This simple principle combined with other ideas appears to be quite powerful for many problems. This present paper is a survey on algorithms in operations research and discrete mathematics using pivots. We give also examples where this principle allows not only to compute but also to prove some theorems in a constructive way. A formalisation is described, mainly based on ideas by Michael J. Todd. 1.