  We present a new methodology for computing approximate Nash equilibria for
two-person non-cooperative games based upon certain extensions and
specializations of an existing optimization approach previously used for the
derivation of fixed approximations for this problem. In particular, the general
two-person problem is reduced to an indefinite quadratic programming problem of
special structure involving the $n \times n$ adjacency matrix of an induced
simple graph specified by the input data of the game, where $n$ is the number
of players' strategies. Using this methodology and exploiting certain
properties of the positive part of the spectrum of the induced graph, we show
that for any $\varepsilon > 0$ there is an algorithm to compute an
$\varepsilon$-approximate Nash equilibrium in time $n^{\xi(m)/\varepsilon}$,
where, $\xi (m) = \sum_{i=1}^m \lambda_i / n$ and $\lambda_1, \lambda_2, >...,
\lambda_m$ are the positive eigenvalues of the adjacency matrix of the graph.
For classes of games for which $\xi (m)$ is a constant, there is a PTAS. Based
on the best upper bound derived for $\xi(m)$ so far, the worst case complexity
of the method is bounded by the subexponential $n^{\sqrt{m}/\varepsilon}$.
