Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?

Spiking Neural Networks Through the Lens of Streaming Algorithms

We initiate the study of biological neural networks from the perspective of streaming algorithms. Like computers, human brains suffer from memory limitations which pose a significant obstacle when processing large scale and dynamically changing data. In computer science, these challenges are captured by the well-known streaming model, which can be traced back to Munro and Paterson `78 and has had significant impact in theory and beyond. In the classical streaming setting, one must compute some function $f$ of a stream of updates $\mathcal{S} = \{u_1,\ldots,u_m\}$, given restricted single-pass access to the stream. The primary complexity measure is the space used by the algorithm. We take the first steps towards understanding the connection between streaming and neural algorithms. On the upper bound side, we design neural algorithms based on known streaming algorithms for fundamental tasks, including distinct elements, approximate median, heavy hitters, and more. The number of neurons in our neural solutions almost matches the space bounds of the corresponding streaming algorithms. As a general algorithmic primitive, we show how to implement the important streaming technique of linear sketching efficient in spiking neural networks. On the lower bound side, we give a generic reduction, showing that any space-efficient spiking neural network can be simulated by a space-efficiently streaming algorithm. This reduction lets us translate streaming-space lower bounds into nearly matching neural-space lower bounds, establishing a close connection between these two models.Comment: To appear in DISC'20, shorten abstrac

Efficient Algorithms for Certifying Lower Bounds on the Discrepancy of Random Matrices

We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix $A \in \mathbb{R}^{m \times n}$, output a value that is a lower bound on $\mathsf{disc}(A) = \min_{x \in \{\pm 1\}^n} ||Ax||_\infty$ for every $A$, but is close to the typical value of $\mathsf{disc}(A)$ with high probability over the choice of a random $A$. This problem is important because of its connections to conjecturally-hard average-case problems such as negatively-spiked PCA, the number-balancing problem and refuting random constraint satisfaction problems. We give the first polynomial-time algorithms with non-trivial guarantees for two main settings. First, when the entries of $A$ are i.i.d. standard Gaussians, it is known that $\mathsf{disc} (A) = \Theta (\sqrt{n}2^{-n/m})$ with high probability. Our algorithm certifies that $\mathsf{disc}(A) \geq \exp(- O(n^2/m))$ with high probability. As an application, this formally refutes a conjecture of Bandeira, Kunisky, and Wein on the computational hardness of the detection problem in the negatively-spiked Wishart model. Second, we consider the integer partitioning problem: given $n$ uniformly random $b$-bit integers $a_1, \ldots, a_n$, certify the non-existence of a perfect partition, i.e. certify that $\mathsf{disc} (A) \geq 1$ for $A = (a_1, \ldots, a_n)$. Under the scaling $b = \alpha n$, it is known that the probability of the existence of a perfect partition undergoes a phase transition from 1 to 0 at $\alpha = 1$; our algorithm certifies the non-existence of perfect partitions for some $\alpha = O(n)$. We also give efficient non-deterministic algorithms with significantly improved guarantees. Our algorithms involve a reduction to the Shortest Vector Problem.Comment: ITCS 202

Individual Fairness in Advertising Auctions Through Inverse Proportionality

Recent empirical work demonstrates that online advertisement can exhibit bias in the delivery of ads across users even when all advertisers bid in a non-discriminatory manner. We study the design ad auctions that, given fair bids, are guaranteed to produce fair outcomes. Following the works of Dwork and Ilvento [2019] and Chawla et al. [2020], our goal is to design a truthful auction that satisfies "individual fairness" in its outcomes: informally speaking, users that are similar to each other should obtain similar allocations of ads. Within this framework we quantify the tradeoff between social welfare maximization and fairness. This work makes two conceptual contributions. First, we express the fairness constraint as a kind of stability condition: any two users that are assigned multiplicatively similar values by all the advertisers must receive additively similar allocations for each advertiser. This value stability constraint is expressed as a function that maps the multiplicative distance between value vectors to the maximum allowable ?_{?} distance between the corresponding allocations. Standard auctions do not satisfy this kind of value stability. Second, we introduce a new class of allocation algorithms called Inverse Proportional Allocation that achieve a near optimal tradeoff between fairness and social welfare for a broad and expressive class of value stability conditions. These allocation algorithms are truthful and prior-free, and achieve a constant factor approximation to the optimal (unconstrained) social welfare. In particular, the approximation ratio is independent of the number of advertisers in the system. In this respect, these allocation algorithms greatly surpass the guarantees achieved in previous work. We also extend our results to broader notions of fairness that we call subset fairness

Robust Algorithms Under Adversarial Injections

In this paper, we study streaming and online algorithms in the context of randomness in the input. For several problems, a random order of the input sequence - as opposed to the worst-case order - appears to be a necessary evil in order to prove satisfying guarantees. However, algorithmic techniques that work under this assumption tend to be vulnerable to even small changes in the distribution. For this reason, we propose a new adversarial injections model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We believe that studying algorithms under this much weaker assumption can lead to new insights and, in particular, more robust algorithms. We investigate two classical combinatorial-optimization problems in this model: Maximum matching and cardinality constrained monotone submodular function maximization. Our main technical contribution is a novel streaming algorithm for the latter that computes a 0.55-approximation. While the algorithm itself is clean and simple, an involved analysis shows that it emulates a subdivision of the input stream which can be used to greatly limit the power of the adversary

Canadian Traveller Problem with Predictions

In this work, we consider the $k$-Canadian Traveller Problem ($k$-CTP) under the learning-augmented framework proposed by Lykouris & Vassilvitskii. $k$-CTP is a generalization of the shortest path problem, and involves a traveller who knows the entire graph in advance and wishes to find the shortest route from a source vertex $s$ to a destination vertex $t$, but discovers online that some edges (up to $k$) are blocked once reaching them. A potentially imperfect predictor gives us the number and the locations of the blocked edges. We present a deterministic and a randomized online algorithm for the learning-augmented $k$-CTP that achieve a tradeoff between consistency (quality of the solution when the prediction is correct) and robustness (quality of the solution when there are errors in the prediction). Moreover, we prove a matching lower bound for the deterministic case establishing that the tradeoff between consistency and robustness is optimal, and show a lower bound for the randomized algorithm. Finally, we prove several deterministic and randomized lower bounds on the competitive ratio of $k$-CTP depending on the prediction error, and complement them, in most cases, with matching upper bounds

Strategyproof Scheduling with Predictions

In their seminal paper that initiated the field of algorithmic mechanism design, Nisan and Ronen [Noam Nisan and Amir Ronen, 1999] studied the problem of designing strategyproof mechanisms for scheduling jobs on unrelated machines aiming to minimize the makespan. They provided a strategyproof mechanism that achieves an n-approximation and they made the bold conjecture that this is the best approximation achievable by any deterministic strategyproof scheduling mechanism. After more than two decades and several efforts, n remains the best known approximation and very recent work by Christodoulou et al. [George Christodoulou et al., 2021] has been able to prove an ?(?n) approximation lower bound for all deterministic strategyproof mechanisms. This strong negative result, however, heavily depends on the fact that the performance of these mechanisms is evaluated using worst-case analysis. To overcome such overly pessimistic, and often uninformative, worst-case bounds, a surge of recent work has focused on the "learning-augmented framework", whose goal is to leverage machine-learned predictions to obtain improved approximations when these predictions are accurate (consistency), while also achieving near-optimal worst-case approximations even when the predictions are arbitrarily wrong (robustness). In this work, we study the classic strategic scheduling problem of Nisan and Ronen [Noam Nisan and Amir Ronen, 1999] using the learning-augmented framework and give a deterministic polynomial-time strategyproof mechanism that is 6-consistent and 2n-robust. We thus achieve the "best of both worlds": an O(1) consistency and an O(n) robustness that asymptotically matches the best-known approximation. We then extend this result to provide more general worst-case approximation guarantees as a function of the prediction error. Finally, we complement our positive results by showing that any 1-consistent deterministic strategyproof mechanism has unbounded robustness

On Constructing Spanners from Random Gaussian Projections

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work