In an ǫ-Nash equilibrium, a player can gain at most ǫ by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in [0, 1], the best-known ǫ achievable in polynomial time is 0.3393 [24]. In general, for n-player games an ǫ-Nash equilibrium can be computed in polynomial time for an ǫ that is an increasing function of n but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of ǫ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general n-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant ǫ such that computing an ǫ-Nash equilibrium of a polymatrix game is PPAD-hard. Our main result is that a (0.5 + δ)-Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and Inspired by the algorithm of Tsaknakis and Spirakis [24], our algorithm uses gradient descent on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a (0.5 + δ)-Nash equilibrium in a two-player Bayesian game.

We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete [28, 29], we show that the exact version is much harder. Specifically, finding a solution with n agents and n cuts is FIXP-hard, and deciding whether there exists a solution with fewer than n cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP ⊆ BU ⊆ TFETR and that LinearBU = PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.

We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem ǫ-NE δ-SW: find an ǫ-approximate Nash equilibrium (ǫ-NE) that is within δ of the best social welfare achievable by an ǫ-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving 1 8 − O(δ) -NE O(δ)-SW for an n × n bimatrix game requires n Ω(log n) time. Building on this result, we show similar conditional running time lower bounds on a number of decision problems for approximate Nash equilibria that do not involve social welfare, including maximizing or minimizing a certain player's payoff, or finding approximate equilibria contained in a given pair of supports. We show quasipolynomial lower bounds for these problems assuming that ETH holds, where these lower bounds apply to ǫ-Nash equilibria for all ǫ < 1 8 . The hardness of these other decision problems has so far only been studied in the context of exact equilibria.

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then compute approximate Nash equilibria using only limited communication between the players. Our method has several applications for improved bounds for efficient computations of approximate Nash equilibria in bimatrix games. First, it yields a best polynomial-time algorithm for computing approximate wellsupported Nash equilibria (WSNE), which guarantees to find a 0.6528-WSNE in polynomial time. Furthermore, since our algorithm solves the two LPs separately, it can be used to improve upon the best known algorithms in the limited communication setting: the algorithm can be implemented to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best known bound in the query complexity setting, requiring O(n log n) payoff queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to provide the best known communication efficient algorithm for computing approximate Nash equilibria: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium.Communication complexity of equilibria in games has been studied in previous works [3,13]. The recent paper of Goldberg and Pastink [11] initiated the study of communication complexity in the bimatrix game setting. There they showed Θ(n 2 ) communication is required to find an exact Nash equilibrium of an n × n bimatrix game. Since these games have Θ(n 2 ) payoffs in total, this implies that there is no communication efficient protocol for finding exact Nash equilibria in bimatrix games. For approximate equilibria, they showed that one can find a 3 4 -Nash equilibrium without any communication, and that in the nocommunication setting, finding an 1 2 -Nash equilibrium is impossible. Motivated by these positive and negative results, they focused on the most interesting setting, which allows only a polylogarithmic (in n) amount of communication (number of bits) between the players. They demonstrated that one can compute 0.438-Nash equilibria and 0.732-well-supported Nash equilibria in this setting.

Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem in computational chemistry. Due to the exponentially large search space, the problem has been referred in several materials-science papers as "NP-Hard and very challenging" without any formal proof though. This paper fills a gap in the literature providing the first set of formally proven NP-Hardness results for a variant of csp with various realistic constraints. In particular, we focus on the problem of removal : the goal is to find a substructure with minimal potential energy, by removing a subset of the ions from a given initial structure. Our main contributions are NP-Hardness results for the csp removal problem, new embeddings of combinatorial graph problems into geometrical settings, and a more systematic exploration of the energy function to reveal the complexity of csp. In a wider context, our results contribute to the analysis of computational problems for weighted graphs embedded into the three-dimensional Euclidean space.