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1. Approximating Cumulative Pebbling Cost Is Unique Games Hard

   https://doi.org/10.4230/LIPIcs.ITCS.2020.13  

   Blocki, J ; Lee, S ; Zhou, S. ( January 2020 , Leibniz international proceedings in informatics)

   null (Ed.)

   The cumulative pebbling complexity of a directed acyclic graph G is defined as cc(G) = min_P ∑_i |P_i|, where the minimum is taken over all legal (parallel) black pebblings of G and |P_i| denotes the number of pebbles on the graph during round i. Intuitively, cc(G) captures the amortized Space-Time complexity of pebbling m copies of G in parallel. The cumulative pebbling complexity of a graph G is of particular interest in the field of cryptography as cc(G) is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) f_{G,H} [Joël Alwen and Vladimir Serbinenko, 2015] defined using a constant indegree directed acyclic graph (DAG) G and a random oracle H(⋅). A secure iMHF should have amortized Space-Time complexity as high as possible, e.g., to deter brute-force password attacker who wants to find x such that f_{G,H}(x) = h. Thus, to analyze the (in)security of a candidate iMHF f_{G,H}, it is crucial to estimate the value cc(G) but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that it is NP-Hard to compute cc(G), but their techniques do not even rule out an efficient (1+ε)-approximation algorithm for any constant ε>0. We show that for any constant c > 0, it is Unique Games hard to approximate cc(G) to within a factor of c. Along the way, we show the hardness of approximation of the DAG Vertex Deletion problem on DAGs of constant indegree. Namely, we show that for any k,ε >0 and given a DAG G with N nodes and constant indegree, it is Unique Games hard to distinguish between the case that G is (e_1, d_1)-reducible with e_1=N^{1/(1+2 ε)}/k and d_1=k N^{2 ε/(1+2 ε)}, and the case that G is (e_2, d_2)-depth-robust with e_2 = (1-ε)k e_1 and d_2= 0.9 N^{(1+ε)/(1+2 ε)}, which may be of independent interest. Our result generalizes a result of Svensson who proved an analogous result for DAGs with indegree 𝒪(N). 

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2. The Quest for Strong Inapproximability Results with Perfect Completeness

   https://doi.org/10.1145/3459668  

   Brakensiek, Joshua ; Guruswami, Venkatesan ( August 2021 , ACM Transactions on Algorithms)

   The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices. 

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3. Two-Sided Kirszbraun Theorem

   https://doi.org/10.4230/LIPIcs.SoCG.2021.13  

   Backurs, Arturs ; Mahabadi, Sepideh ; Makarychev, Konstantin ; Makarychev, Yury ( June 2021 , Leibniz international proceedings in informatics)

   Buchin, Kevin ; Colin de Verdiere, Eric (Ed.)

   In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖. 

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4. Model-Free Reinforcement Learning for Stochastic Parity Games

   Hahn, Ernst Moritz ; Perez, Mateo ; Schewe, Sven ; Somenzi, Fabio ; Trivedi, Ashutosh ; Wojtczak, Dominik ( August 2020 , 31st International Conference on Concurrency Theory (CONCUR 2020))

   null (Ed.)

   This paper investigates the use of model-free reinforcement learning to compute the optimal value in two-player stochastic games with parity objectives. In this setting, two decision makers, player Min and player Max, compete on a finite game arena - a stochastic game graph with unknown but fixed probability distributions - to minimize and maximize, respectively, the probability of satisfying a parity objective. We give a reduction from stochastic parity games to a family of stochastic reachability games with a parameter ε, such that the value of a stochastic parity game equals the limit of the values of the corresponding simple stochastic games as the parameter ε tends to 0. Since this reduction does not require the knowledge of the probabilistic transition structure of the underlying game arena, model-free reinforcement learning algorithms, such as minimax Q-learning, can be used to approximate the value and mutual best-response strategies for both players in the underlying stochastic parity game. We also present a streamlined reduction from 1 1/2-player parity games to reachability games that avoids recourse to nondeterminism. Finally, we report on the experimental evaluations of both reductions 

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5. Deep fictitious play for stochastic differential games

   https://doi.org/https://dx.doi.org/10.4310/CMS.2021.v19.n2.a2  

   Hu, Ruimeng ( January 2021 , Communications in mathematical sciences)

   Jin, Shi (Ed.)

   In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric N-player non-zero-sum stochastic differential games, for which we refer as deep fictitious play, a multi-stage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize her own payoff subject to the other players’ previous actions, equivalent to solving N decoupled stochastic control optimization problems, which are approximated by DNNs. Therefore, the fictitious play strategy leads to a structure consisting of N DNNs, which only communicate at the end of each stage. The resulting deep learning algorithm based on fictitious play is scalable, parallel and model-free, i.e., using GPU parallelization, it can be applied to any N-player stochastic differential game with different symmetries and heterogeneities (e.g., existence of major players). We illustrate the performance of the deep learning algorithm by comparing to the closed-form solution of the linear quadratic game. Moreover, we prove the convergence of fictitious play under appropriate assumptions, and verify that the convergent limit forms an open-loop Nash equilibrium. We also discuss the extensions to other strategies designed upon fictitious play and closed-loop Nash equilibrium in the end. 

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