Approximating Nash Equilibria and Dense Subgraphs via an Approximate Version of Carath\'{e}odory's Theorem

We present algorithmic applications of an approximate version of Carath\'{e}odory's theorem. The theorem states that given a set of vectors $X$ in $\mathbb{R}^d$, for every vector in the convex hull of $X$ there exists an $\varepsilon$-close (under the $p$-norm distance, for $2\leq p < \infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ depends on $\varepsilon$ and the norm $p$ and is independent of the dimension $d$. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with $n \times n$ payoff matrices $A, B$, if the number of non-zero entries in any column of $A+B$ is at most $s$ then an $\varepsilon$-Nash equilibrium of the game can be computed in time $n^{O\left(\frac{\log s }{\varepsilon^2}\right)}$. This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity $s$. Moreover, for arbitrary bimatrix games---since $s$ can be at most $n$---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carath\'{e}odory's theorem also leads to an additive approximation algorithm for the normalized densest $k$-subgraph problem. Given a graph with $n$ vertices and maximum degree $d$, the developed algorithm determines a subgraph with exactly $k$ vertices with normalized density within $\varepsilon$ (in the additive sense) of the optimal in time $n^{O\left( \frac{\log d}{\varepsilon^2}\right)}$. Additionally, we show that a similar approximation result can be achieved for the problem of finding a $k \times k$-bipartite subgraph of maximum normalized density.