Aviad Rubinstein

Settling the complexity of computing approximate two-player Nash equilibria

We prove that there exists a constant ε > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an ε-approximate Nash equilibrium in a two-player (n × n) game requires quasi-polynomial time, nlog1-o(1) n. This matches (up to the o(1) term) the algorithm of Lipton, Markakis, and Mehta [54]. Our proof relies on a variety of techniques from the study of probabilistically checkable proofs (PCP), this is the first time that such ideas are used for a reduction between problems inside PPAD. En route, we also prove new hardness results for computing Nash equilibria in games with many players. In particular, we show that computing an ε-approximate Nash equilibrium in a game with n players requires 2Ω(n) oracle queries to the payoff tensors. This resolves an open problem posed by Hart and Nisan [43], Babichenko [13], and Chen et al. [28]. In fact, our results for n-player games are stronger: they hold with respect to the (ε,δ)-WeakNash relaxation recently introduced by Babichenko et al. [15].

The complexity of fairness through equilibrium

Competitive equilibrium with equal incomes (CEEI) is a well-known fair allocation mechanism [Foley67:Resource, Varian74: Equity, Thomson85:Theories]; however, for indivisible resources a CEEI may not exist. It was shown in Budish [2011] that in the case of indivisible resources there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient, for some favorable approximation parameters. This approximation is used in practice to assign business school students to classes. In this paper we show that finding the A-CEEI allocation guaranteed to exist by Budishs theorem is PPAD-complete. We further show that finding an approximate equilibrium with better approximation guarantees is even harder: NP-complete.

On the Computational Complexity of Optimal Simple Mechanisms

We consider a monopolist seller facing a single buyer with additive valuations over n heterogeneous, independent items. It is known that in this important setting optimal mechanisms may require randomization [12], use menus of infinite size [9], and may be computationally intractable [8]. This has sparked recent interest in finding simple mechanisms that obtain reasonable approximations to the optimal revenue [10, 15, 3]. In this work we attempt to find the optimal simple mechanism. There are many ways to define simple mechanisms. Here we restrict our search to partition mechanisms, where the seller partitions the items into disjoint bundles and posts a price for each bundle; the buyer is allowed to buy any number of bundles. We give a PTAS for the problem of finding a revenue-maximizing partition mechanism, and prove that the problem is strongly NP-hard. En route, we prove structural properties of near-optimal partition mechanisms which may be of independent interest: for example, there always exists a near-optimal partition mechanism that uses only a constant number of non-trivial bundles (i.e. bundles with more than one item).

Settling the Complexity of Computing Approximate Two-Player Nash Equilibria

We prove that there exists a constant e > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an e-approximate Nash equilibrium in a two-player (n × n) game requires quasi-polynomial time, nlog1-o(1) n. This matches (up to the o(1) term) the algorithm of Lipton, Markakis, and Mehta [54]. Our proof relies on a variety of techniques from the study of probabilistically checkable proofs (PCP), this is the first time that such ideas are used for a reduction between problems inside PPAD. En route, we also prove new hardness results for computing Nash equilibria in games with many players. In particular, we show that computing an e-approximate Nash equilibrium in a game with n players requires 2Ω(n) oracle queries to the payoff tensors. This resolves an open problem posed by Hart and Nisan [43], Babichenko [13], and Chen et al. [28]. In fact, our results for n-player games are stronger: they hold with respect to the (e,δ)-WeakNash relaxation recently introduced by Babichenko et al. [15].

Communication complexity of approximate Nash equilibria

For a constant ϵ, we prove a (N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player N x N games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1-ϵ)-fraction of the players are ϵ-best replying.

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed} model of probabilistically checkable proofs (PCP). A satisfying assignment x ∊ \{0,1\}^n to a CNF formula \phi is shared between two parties, where Alice knows x_1, \dots, x_{n/2, Bob knows x_{n/2+1},\dots,x_n, and both parties know \phi. The goal is to have Alice and Bob jointly write a PCP that x satisfies \phi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic} variant, where the players are helped by Merlin, a third party who knows all of x.Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in \P. In particular, under SETH we show that %(assuming SETH) there are no truly-subquadratic approximation algorithms for %the following problems: Maximum Inner Product over \{0,1\}-vectors, LCS Closest Pair over permutations, Approximate Partial Match, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first three problems we obtain nearly-polynomial factors of 2^{(log n)^{1-o(1)}};only (1+o(1))-factor lower bounds (under SETH) were known before.As an additional feature of our reduction, we obtain new SETH lower bounds for the exact} monochromatic Closest Pair problem in the Euclidean, Manhattan, and Hamming metrics.

Locally adaptive optimization: adaptive seeding for monotone submodular functions

The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize. Our main result is a (1 -- 1/e)2-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call locally-adaptive policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the (1 -- 1/e)2-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies. We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most k that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from P lanted -C lique that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of locally-adaptive policies we use in the main result.

Combining Traditional Marketing and Viral Marketing with Amphibious Influence Maximization

In this paper, we propose the amphibious influence maximization (AIM) model that combines traditional marketing via content providers and viral marketing to consumers in social networks in a single framework. In AIM, a set of content providers and consumers form a bipartite network while consumers also form their social network, and influence propagates from the content providers to consumers and among consumers in the social network following the independent cascade model. An advertiser needs to select a subset of seed content providers and a subset of seed consumers, such that the influence from the seed providers passing through the seed consumers could reach a large number of consumers in the social network in expectation. We prove that the AIM problem is NP-hard to approximate to within any constant factor via a reduction from Feiges k-prover proof system for 3-SAT5. We also give evidence that even when the social network graph is trivial (i.e. has no edges), a polynomial time constant factor approximation for AIM is unlikely. However, when we assume that the weighted bi-adjacency matrix that describes the influence of content providers on consumers is of constant rank, a common assumption often used in recommender systems, we provide a polynomial-time algorithm that achieves approximation ratio of (1-1/e-ε)3 for any (polynomially small) ε > 0. Our algorithmic results still hold for a more general model where cascades in social network follow a general monotone and submodular function.

Combinatorial prophet inequalities

We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward-closed feasibility constraint, we give an O(log n log2 r)-competitive algorithm (where r is the cardinality of the largest feasible subset). Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an [EQUATION] lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model. En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for non-monotone submodular functions.

The limitations of optimization from samples

In this paper we consider the following question: can we optimize objective functions from the training data we use to learn them? We formalize this question through a novel framework we call optimization from samples (OPS). In OPS, we are given sampled values of a function drawn from some distribution and the objective is to optimize the function under some constraint. While there are interesting classes of functions that can be optimized from samples, our main result is an impossibility. We show that there are classes of functions which are statistically learnable and optimizable, but for which no reasonable approximation for optimization from samples is achievable. In particular, our main result shows that there is no constant factor approximation for maximizing coverage functions under a cardinality constraint using polynomially-many samples drawn from any distribution. We also show tight approximation guarantees for maximization under a cardinality constraint of several interesting classes of functions including unit-demand, additive, and general monotone submodular functions, as well as a constant factor approximation for monotone submodular functions with bounded curvature.