APPROXIMATING NASH EQUILIBRIA AND DENSE SUBGRAPHS VIA AN APPROXIMATE VERSION OF CARATHEODORY'S THEOREM

摘要

We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in R-d, for every vector in the convex hull of X there exists an epsilon-close (under the p-norm distance for 2 = p infinity) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on epsilon 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 x n payoff matrices A, B, if the number of nonzero entries in any column of A + B is at most s then an epsilon-Nash equilibrium of the game can be computed in time n(O(log s/epsilon 2)). 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 the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The 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, our algorithm determines a size-k subgraph with normalized density within epsilon of the optimal in time n(O(log s/epsilon 2)).