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 et al. [2006a] on the complexity of four-player Nash equilibria, 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.

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We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time.Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges, our algorithm computes a (1− )-approximately maximum s-t flow in time 1 O mn 1/3 −11/3 . A dual version of our approach computes a (1 + )-approximately minimum s-t cut in time O m + n 4/3 −8/3 , which is the fastest known algorithm for this problem as well. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum s-t flows in time O m √ n −1 , and approximately minimum s-t cuts in time O m + n 3/2 −3 .

A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system ⌫, there is a sphere S that intersects at most O(k 1/d n 1Ϫ1/d) balls of ⌫ and divides the remainder of ⌫ into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 Ϫ 1/(d ϩ 2))n balls. This bound of O(k 1/d n 1Ϫ1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every k-nearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1Ϫ1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

Spectral partitioning methods use the Fiedler vector|the eigenvector of the secondsmallest eigenvalue of the Laplacian matrix|to nd a small separator of a graph. These methods are important components of many scienti c numerical algorithms and have been demonstrated by experiment t o w ork extremely well. In this paper, we show that spectral partitioning methods work well on bounded-degree planar graphs and nite element meshes| the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O p n for bounded-degree planar graphs and two-dimensional meshes and O n 1=d for well-shaped d-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs.

We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster-a subset of vertices whose internal connections are significantly richer than its external connectionsnear a given vertex. The running time of our algorithm, when it finds a non-empty local cluster, is nearly linear in the size of the cluster it outputs.Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and web-graphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph.Using the partitioning algorithm of this paper, we have designed a nearly-linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearlylinear time algorithm for solving linear systems in symmetric, diagonally-dominant matrices. The linear system solver also leads to a nearly linear-time algorithm for approximating the second-smallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers. * This paper is the first in a sequence of three papers expanding on material that appeared first under the title "Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems" [ST03]. The second paper, "Spectral Sparsification of Graphs" [ST08b] contains further results on partitioning graphs, and applies them to producing spectral sparsifiers of graphs. The third paper, "Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems" [ST08a] contains the results on solving linear equations and approximating eigenvalues and eigenvectors.

Spectral partitioning methods use the Fiedler vectorthe eigenvector of the second-smallest eigenvalue of the Laplacian matrix-to find a small separator of a graph. These methods are important components of m a n y scientific numerical algorithms and have been demonstrated by experzment to work extremely well. In this paper, we show that spectral partitioning methods work well o n bounded-degree planar graphs and finite element meshesthe classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O ( m f o r bounded-degree planar graphs and two-dimensional meshes and O(n'ld) f o r well-shaped ddimensional meshes. The heart of our analysis is a n upper bound o n the second-smallest eigenvalues of the Laplacian matrices of these graphs: we prove a bound of O(l/n) for bounded-degree planar graphs and O ( l / n 2 / d ) for wellshaped d-dimensional meshes.