We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games.For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an Erdős-Rényi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets.For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) 4 3 − ǫ -approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a 4 3 -approximation for linear latencies.

We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced k-clique and a graph in which all k-subgraphs have density at most 1−ε, requires nΩ (log n) time. Our result essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and is the first one to rule out an additive PTAS for Densest k-Subgraph. We further strengthen this result by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by a near-polynomial factor (k = k · 2 −Ω(log n) ) are assumed to be at most (1 − ε)-dense.Our reduction is inspired by recent applications of the "birthday repetition" technique [AIM14, BKW15]. Our analysis relies on information theoretical machinery and is similar in spirit to analyzing a parallel repetition of twoprover games in which the provers may choose to answer some challenges multiple times, while completely ignoring other challenges.

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We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Ω(n/r + r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was Ω(n/r 2 + r) due to Jain, Radhakrishnan and Sen [JRS03]. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function f is at most 2 O(QIC(f )) , where QIC(f ) is the prior-free quantum information complexity of f (with error 1/3). Research supported in part by an FRQNT B2 Doctoral Research Scholarship and by CryptoWorks21.6 From AND to Disj 31 7 Proof of the main result 34 8 Low information protocol for AND 35 2 1 Introduction We prove near-optimal bounds on the bounded-round quantum communication complexity of disjointness. Quantum communication complexity, introduced by Yao [Yao93], studies the amount of quantum communication that two parties, Alice and Bob, need to exchange in order to compute a function (usually boolean) of their private inputs. It is the natural quantum extension of classical communication complexity [Yao79]. While the inputs are classical and the end result is classical, the players are allowed to use quantum resources while communicating. The motivation for the introduction of quantum communication was to study questions in quantum computation. For example, in [Yao93], Yao used it to prove that the majority function does not have any linear size quantum formulas. While quantum communication (with entanglement) offers only a factor of 2 savings when transmitting n bits of classical information [Hol73, BW92, CvDNT98], it can still offer superconstant savings (and sometimes exponential) in communication if the goal is just to compute a boolean function of the inputs. For total boolean functions, the best-known separation between classical and quantum communication is quadratic, for the disjointness function [KS92, Raz92, Gro96, BCW98, AA03]. It is, in fact, a major open problem whether classical and quantum communication are polynomially related for all total boolean functions. For partial functions, exponential separations are known even between one-way quantum communication and arbitrary classical communication [Raz99, KR11]. For disjointness with input size n, Grover's search [Gro96, BBHT98] can be used to obtain a quantum communication protocol (with probability of error 1/3) with communication cost O( √ n log n) [BCW98]. The bound was later improved to O( √ n) in [AA03]. The protocols attaining 2 Proof overview and discussion High-level strategy. At a high-level, the proof builds on the connection between quantum information complexity and quantum communication complexity of the disjointness function DISJ m with various values of m. There ar...

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