We study the $\eps$-rank of a real matrix $A$, defined for any $\eps > 0$ as the minimum rank of a matrix that approximates every entry of $A$ to within an additive $\eps$. This parameter is connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and learning theory. Here we give bounds on the $\eps$-rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive approximation schemes for Nash equilibria for $2$-player games when the payoff matrices are positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted graphs. We use combinatorial, geometric and spectral techniques; our main new tool is an efficient algorithm for the following problem: given a convex body $A$ and a symmetric convex body $B$, find a covering a $A$ with translates of $B$.

Modified enumeration algorithm. Corrected errors. Made algorithms deterministic.

We study the $\eps$-rank of a real matrix $A$, defined for any $\eps > 0$ as the minimum rank over matrices that approximate every entry of $A$ to within an additive $\eps$. This parameter is connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and learning theory. Here we give bounds on the $\eps$-rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive
approximation schemes for Nash equilibria for $2$-player games when the payoff matrices are
positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted graphs. We use combinatorial, geometric and spectral techniques; our main new tool is an algorithm for efficiently covering a convex body with translates of another convex body.

Added references to work in communication complexity; added two new co-authors.

We introduce and study the \epsilon-rank of a real matrix A, defined, for any  \epsilon > 0 as the minimum rank over matrices that approximate every entry of A to within an additive \epsilon. This parameter is connected to other notions of approximate rank and is motivated by problems from various topics including combinatorial optimization, game theory, computational geometry and learning theory. Here we give bounds on the \epsilon-rank and use them to derive (a) polynomial-time approximation schemes for Nash equilibria of substantially larger classes of 2-player games than previously known and (b) an additive PTAS for the densest subgraph problem on inputs having small \epsilon-rank. We use combinatorial, geometric and spectral techniques; our main new tool is an algorithm for efficiently covering a convex body with translates of another convex body.