Computational Complexity of Approximate Nash Equilibrium in Large Games

We prove that finding an epsilon-Nash equilibrium in a succinctly representable game with many players is PPAD-hard for constant epsilon. Our proof uses succinct games, i.e. games whose payoff function is represented by a circuit. Our techniques build on a recent query complexity lower bound by Babichenko.Comment: New version includes an addendum about subsequent work on the open problems propose

Honest signaling in zero-sum games is hard, and lying is even harder

We prove that, assuming the exponential time hypothesis, finding an \epsilon-approximately optimal symmetric signaling scheme in a two-player zero-sum game requires quasi-polynomial time. This is tight by [Cheng et al., FOCS'15] and resolves an open question of [Dughmi, FOCS'14]. We also prove that finding a multiplicative approximation is NP-hard. We also introduce a new model where a dishonest signaler may publicly commit to use one scheme, but post signals according to a different scheme. For this model, we prove that even finding a (1-2^{-n})-approximately optimal scheme is NP-hard

Constant-factor approximation of near-linear edit distance in near-linear time

We show that the edit distance between two strings of length $n$ can be computed within a factor of $f(\epsilon)$ in $n^{1+\epsilon}$ time as long as the edit distance is at least $n^{1-\delta}$ for some $\delta(\epsilon) > 0$.Comment: 40 pages, 4 figure

Detecting communities is Hard (And Counting Them is Even Harder)

We consider the algorithmic problem of community detection in networks. Given an undirected friendship graph G, a subset S of vertices is an (a,b)-community if: * Every member of the community is friends with an (a)-fraction of the community; and * every non-member is friends with at most a (b)-fraction of the community. [Arora, Ge, Sachdeva, Schoenebeck 2012] gave a quasi-polynomial time algorithm for enumerating all the (a,b)-communities for any constants a>b. Here, we prove that, assuming the Exponential Time Hypothesis (ETH), quasi-polynomial time is in fact necessary - and even for a much weaker approximation desideratum. Namely, distinguishing between: * G contains an (1,o(1))-community; and * G does not contain a (b,b+o(1))-community for any b. We also prove that counting the number of (1,o(1))-communities requires quasi-polynomial time assuming the weaker #ETH