Journal of the ACM最新文献

Bounding the price of anarchy, which quantifies the damage to social welfare due to selfish behavior of the participants, has been an important area of research in algorithmic game theory. Classical work on such bounds in repeated games makes the strong assumption that the subsequent rounds of the repeated games are independent beyond any influence on play from past history. This work studies such bounds in environments that themselves change due to the actions of the agents. Concretely, we consider this problem in discrete-time queuing systems, where competitive queues try to get their packets served. In this model, a queue gets to send a packet at each step to one of the servers, which will attempt to serve the oldest arriving packet, and unprocessed packets are returned to each queue. We model this as a repeated game where queues compete for the capacity of the servers, but where the state of the game evolves as the length of each queue varies.

We analyze this queuing system from multiple perspectives. As a baseline measure, we first establish precise conditions on the queuing arrival rates and service capacities that ensure all packets clear efficiently under centralized coordination. We then show that if queues strategically choose servers according to independent and stationary distributions, the system remains stable provided it would be stable under coordination with arrival rates scaled up by a factor of just (frac{e}{e-1}). Finally, we extend these results to no-regret learning dynamics: if queues use learning algorithms satisfying the no-regret property to choose servers, then the requisite factor increases to 2, and both of these bounds are tight. Both of these results require new probabilistic techniques compared to the classical price of anarchy literature and show that in such settings, no-regret learning can exhibit efficiency loss due to myopia.

This work represents a natural coalescence of two important lines of work — learning mixtures of Gaussians and algorithmic robust statistics. In particular, we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving a type of dimension-independent polynomial identifiability — which we call robust identifiability — through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation.

We consider two-player stochastic games played on a finite graph for infinitely many rounds. Stochastic games generalize both Markov decision processes (MDP) by adding an adversary player, and two-player deterministic games by adding stochasticity. The outcome of the game is a sequence of distributions over the graph states, representing the evolution of a population consisting of a continuum number of identical copies of a process modeled by the game graph. We consider synchronization objectives, which require the probability mass to accumulate in a set of target states, either always, once, infinitely often, or always after some point in the outcome sequence; and the winning modes of sure winning (if the accumulated probability is equal to 1) and almost-sure winning (if the accumulated probability is arbitrarily close to 1).

We present algorithms to compute the set of winning distributions for each of these synchronization modes, showing that the corresponding decision problem is PSPACE-complete for synchronizing once and infinitely often and PTIME-complete for synchronizing always and always after some point. These bounds are remarkably in line with the special case of MDPs, while the algorithmic solution and proof technique are considerably more involved, even for deterministic games. This is because those games have a flavor of imperfect information, in particular they are not determined and randomized strategies need to be considered, even if there is no stochastic choice in the game graph. Moreover, in combination with stochasticity in the game graph, finite-memory strategies are not sufficient in general.

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting.

Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets, i.e., sets respecting the cardinality constraint, are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the above-mentioned link to the (non-streaming) robust setting, both of these algorithms improve on the current state-of-the-art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2 + ε hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.

Abraham, Dolev, Geffner, and Halpern [1] proved that, in asynchronous systems, a (k, t)-robust equilibrium for n players and a trusted mediator can be implemented without the mediator as long as n > 4(k+t), where an equilibrium is (k, t)-robust if, roughly speaking, no coalition of t players can decrease the payoff of any of the other players, and no coalition of k players can increase their payoff by deviating. We prove that this bound is tight, in the sense that if n ≤ 4(k+t) there exist (k, t)-robust equilibria with a mediator that cannot be implemented by the players alone. Even though implementing (k, t)-robust mediators seems closely related to implementing asynchronous multiparty (k+t)-secure computation [6], to the best of our knowledge there is no known straightforward reduction from one problem to another. Nevertheless, we show that there is a non-trivial reduction from a slightly weaker notion of (k+t)-secure computation, which we call (k+t)-strict secure computation, to implementing (k, t)-robust mediators. We prove the desired lower bound by showing that there are functions on n variables that cannot be (k+t)-strictly securely computed if n ≤ 4(k+t). This also provides a simple alternative proof for the well-known lower bound of 4t+1 on asynchronous secure computation in the presence of up to t malicious agents [4, 8, 10].

In the search for a logic capturing polynomial time the most promising candidates are Choiceless Polynomial Time (CPT) and rank logic. Rank logic extends fixed-point logic with counting by a rank operator over prime fields. We show that the isomorphism problem for CFI graphs over ℤ_2ⁱ cannot be defined in rank logic, even if the base graph is totally ordered. However, CPT can define this isomorphism problem. We thereby separate rank logic from CPT and in particular from polynomial time.