Small Clique Detection and Approximate Nash Equilibria

Recently, Hazan and Krauthgamer showed [12] that if, for a fixed small ε, an ε-best ε-approximate Nash equilibrium can be found in polynomial time in two-player games, then it is also possible to find a planted clique in Gn,1/2 of size C log n, where C is a large fixed constant independent of ε. In this paper, we extend their result to show that if an ε-best ε-approximate equilibrium can be efficiently found for arbitrarily small ε > 0, then one can detect the presence of a planted clique of size (2+δ) log n in Gn,1/2 in polynomial time for arbitrarily small δ > 0. Our result is optimal in the sense that graphs in Gn,1/2 have cliques of size (2 − o(1)) log n with high probability.