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

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distribution D. We show that if D is a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who are k-demand, additive up to a matroid constraint, or additive up to constraints of any downwards-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers [LY13, BILW14]. In the second part of the paper, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon that sometimes strictly increasing buyers' values for every set can strictly decrease the revenue of the optimal mechanism [HR12]. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof of approximate revenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms.Comment: Updated title and body to version included in TEAC. Adapted Theorem 5.2 to accommodate \eta-BIC auctions (versus exactly BIC

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