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 ?

Non-Malleable Codes for Small-Depth Circuits

We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e. $\mathsf{AC^0}$ tampering functions), our codes have codeword length $n = k^{1+o(1)}$ for a $k$-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length $2^{O(\sqrt{k})}$. Our construction remains efficient for circuit depths as large as $\Theta(\log(n)/\log\log(n))$ (indeed, our codeword length remains $n\leq k^{1+\epsilon})$, and extending our result beyond this would require separating $\mathsf{P}$ from $\mathsf{NC^1}$. We obtain our codes via a new efficient non-malleable reduction from small-depth tampering to split-state tampering. A novel aspect of our work is the incorporation of techniques from unconditional derandomization into the framework of non-malleable reductions. In particular, a key ingredient in our analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC 2013), a derandomization of the influential switching lemma from circuit complexity; the randomness-efficiency of this switching lemma translates into the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure

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

Learning Reserve Prices in Second-Price Auctions

This paper proves the tight sample complexity of Second-Price Auction with Anonymous Reserve, up to a logarithmic factor, for all value distribution families that have been considered in the literature. Compared to Myerson Auction, whose sample complexity was settled very recently in (Guo, Huang and Zhang, STOC 2019), Anonymous Reserve requires much fewer samples for learning. We follow a similar framework as the Guo-Huang-Zhang work, but replace their information theoretical argument with a direct proof