We advance significantly beyond the recent progress on the algorithmic complexity of Nash equilibria by solving two major open problems in the approximation of Nash equilibria and in the smoothed analysis of algorithms. • We show that no algorithm with complexity poly(n, 1 ɛ) can compute an ɛ-approximate Nash equilibrium in a two-player game, in which each player has n pure strategies, unless PPAD ⊆ P. In other words, the problem of computing a Nash equilibrium in a twoplayer game does not have a fully polynomial-time approximation scheme unless PPAD ⊆ P. • We prove that no algorithm for computing a Nash equilibrium in a two-player game can have smoothed complexity poly(n, 1 σ) under input perturbation of magnitude σ, unless PPAD ⊆ RP. In particular, the smoothed complexity of the classic Lemke-Howson algorithm is not polynomial unless PPAD ⊆ RP. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional hypergrid with constant side-length, and show that it can host the embedding of the proof structure of any PPAD problem. We prove a key geometric lemma for finding a discrete fixed-point, a new concept defined on n +1vertices of a unit hypercube. This lemma enables us to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions. 1

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player 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 four-player 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 two-player Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic Lemke-Howson 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: • Arrow-Debreu market equilibria are PPAD-hard to compute.

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

Abstract. This is the 26th edition of a column that covers new developments in the theory of NP-completeness. 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 NP-Completeness, ” 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