A carbon-free dissemination of ideas across the globe.

### TCS+ talk: Wednesday, May 15th — Ewin Tang, University of Washington

The next TCS+ talk will take place this coming Wednesday, May 15th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Ewin Tang from University of Washington will tell us about “Quantum-inspired classical linear algebra algorithms: why and how?” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: Over the past ten years, the field of quantum machine learning (QML) has produced many polylogarithmic-time procedures for linear algebra routines, assuming certain “state preparation” assumptions. Though such algorithms are formally incomparable with classical computing, a recent line of work uses an analogous classical model of computation as an effective point of comparison to reveal speedups (or lack thereof) gained by QML. The resulting “dequantized” algorithms assume sampling access to input to speed up runtimes to polylogarithmic in input size.

In this talk, we will discuss the motivation behind this model and its relation to existing randomized linear algebra literature. Then, we will delve into an example quantum-inspired algorithm: Gilyen, Lloyd, and Tang’s algorithm for low-rank matrix inversion. This dequantizes a variant of Harrow, Hassidim, and Lloyd’s matrix inversion algorithm, a seminal work in QML. Finally, we will consider the implications of this work on exponential speedups in QML. No background of quantum computing is assumed for this talk.

### TCS+ talk: Wednesday, May 1st — Chris Peikert, University of Michigan

The next TCS+ talk will take place this coming Wednesday, May 1st at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Chris Peikert from University of Michigan will speak about “Noninteractive Zero Knowledge for NP from Learning With Errors” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: We finally close the long-standing problem of constructing a noninteractive zero-knowledge (NIZK) proof system for any NP language with security based on the Learning With Errors (LWE) problem, and thereby on worst-case lattice problems. Our proof system instantiates a framework developed in a series of recent works for soundly applying the Fiat—Shamir transform using a hash function family that is correlation intractable for a suitable class of relations. Previously, such hash families were based either on “exotic” assumptions (e.g., indistinguishability obfuscation or optimal hardness of ad-hoc LWE variants) or, more recently, on the existence of circularly secure fully homomorphic encryption. However, none of these assumptions are known to be implied by LWE or worst-case hardness.

Our main technical contribution is a hash family that is correlation intractable for arbitrary size-$S$ circuits, for any polynomially bounded $S$, based on LWE (with small polynomial approximation factors). Our construction can be instantiated in two possible “modes,” yielding a NIZK that is either computationally sound and statistically zero knowledge in the common random string model, or vice-versa in the common reference string model.

(This is joint work with Sina Shiehian. Paper: https://eprint.iacr.org/2019/158)

### TCS+ talk: Wednesday, April 17th — Thatchaphol Saranurak, TTIC

The next TCS+ talk will take place this coming Wednesday, April 17th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Thatchaphol Saranurak from TTI Chicago will speak about “Breaking Quadratic Time for Small Vertex Connectivity ” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: Vertex connectivity is a classic extensively-studied problem. Given an integer k, its goal is to decide if an $n$-node $m$-edge graph can be disconnected by removing $k$ vertices. Although a $O(m)$-time algorithm was postulated since 1974 [Aho, Hopcroft, and Ullman], and despite its sibling problem of edge connectivity being resolved over two decades ago [Karger STOC’96], so far no vertex connectivity algorithms are faster than $O(n^2)$ time even for $k = 4$ and $m = O(n)$. In the simplest case where $m = O(n)$ and $k = O(1)$, the $O(n^2)$ bound dates five decades back to [Kleitman IEEE Trans. Circuit Theory’69]. For the general case, the best bound is $\tilde{O}( \min{ kn^2, n^\omega + nk^\omega } )$ [Henzinger, Rao, Gabow FOCS’96; Linial, Lovász, Wigderson FOCS’86].

In this talk, I will present a randomized Monte Carlo algorithm with $\tilde{O}(m + k^3 n)$ time. This algorithm proves the conjecture by Aho, Hopcroft, and Ullman when $k=O(1)$ up to a polylog factor, breaks the 50-year-old bound by Kleitman, is fastest for $4 < k < n^{0.456}$. The story is similar for the directed graphs where we obtain an algorithm running time at most $\tilde{O}(k^2 m)$.

The key to our results is to avoid computing single-source connectivity, which was needed by all previous exact algorithms and is not known to admit $o(n^2)$ time. Instead, we design the first local algorithm for computing vertex connectivity; without reading the whole graph, our algorithm can find a separator of size at most $k$ or certify that there is no separator of size at most $k$ “near” a given seed node.

This talk is based on joint works with Danupon Nanongkai and Sorrachai Yingchareonthawornchai.

### TCS+ talk: Wednesday, April 3rd — Richard Peng, Georgia Tech

The next TCS+ talk will take place this coming Wednesday, April 3rd at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Richard Peng from Georgia Tech will speak about “Fully Dynamic Spectral Vertex Sparsifiers and Applications” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: Problems arising from analyzing and understanding graph structures have motivated the development of many powerful tools for storing and compressing graphs and networks. Many, if not most, of these massive graphs are accumulations of updates over time. Maintaining updates and queries on dynamically changing graphs more efficiently than recomputing from scratch is a well-studied topic, with several important open problems related to global, optimization related queries.

We study dynamic graph algorithms for maintaining spectral vertex sparsifiers with respect to a subset of terminal vertices. Such sparsifiers preserve key structures in spectral algorithms, including effective resistances (which can be viewed as a numerical generalization of connectivity), solutions to systems of Laplacian linear equations, and energy of electrical flows between terminal vertices. We give data structures that maintain, in sublinear time, these sparsifiers under insertions/deletions of edges, as well as terminal additions.

This primitive in turn leads to sublinear time data structures for key primitives in spectral graph algorithms, including ones at the core of the “Laplacian paradigm” for designing graph optimization algorithms. In particular, we obtain $O(m^{4/3}\epsilon^{-4})$ time per query/update for effective resistances on unweighted graphs, and $O(n^{11/12}\epsilon^{-5})$ time per query/update for implicit access to linear system solutions, where $\epsilon$ is the relative approximation accuracy.

The majority of the talk will focus on key components of this data structure: (1) an interpretation of vertex sparsifiers as a sum of random walks, (2) a suitable choice of terminals to keep these walks local, and (3) maintenance of local walks and numerical solutions. Potential avenues in generalizing these techniques to provide new building blocks for dynamic graph algorithms will also be discussed.

### TCS+ talk: Wednesday, March 20th, Aleksandar Nikolov, University of Toronto

The next TCS+ talk will take place this coming Wednesday, March 20th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 18:00 Central European Time, 17:00 UTC). Aleksandar Nikolov from University of Toronto will speak about “Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, like Max Cut, for many others, like Max Sat, Max DiCut, and constraint satisfaction problems with global constraints, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known.

In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max Cut, Max 2-Sat, and Max DiCut, and derive new algorithms that are competitive with the best known results. We further show that our algorithms can be used, together with the Sum of Squares hierarchy, to approximate constraint satisfaction problems subject to multiple global cardinality constraints.

Join work with Sepehr Abbasi-Zadeh, Nikhil Bansal, Guru Guruganesh, Roy Schwartz, and Mohit Singh

### TCS+ talk: Wednesday, March 6th, Shayan Oveis Gharan, University of Washington

The next TCS+ talk will take place this coming Wednesday, March 6th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Shayan Oveis Gharan from University of Washington will speak about “Strongly log concave polynomials, high dimensional simplicial complexes, and an FPRAS for counting Bases of Matroids” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: A matroid is an abstract combinatorial object which generalizes the notions of spanning trees, and linearly independent sets of vectors. I will talk about an efficient algorithm based on the Markov Chain Monte Carlo technique to approximately count the number of bases of any given matroid.

The proof is based on a new connections between high dimensional simplicial complexes, and a new class of multivariate polynomials called completely log-concave polynomials. In particular, we exploit a fundamental fact from our previous work that the bases generating polynomial of any given matroid is a log-concave function over the positive orthant.

Based on joint works with Nima Anari, Kuikui Liu, and Cynthia Vinzant.

### TCS+ talk: Wednesday, February 20th, Sepehr Assadi, Princeton

The next TCS+ talk will take place this coming Wednesday, February 20th at
1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European
Time, 18:00 UTC). Sepehr Assadi from Princeton University will speak about “A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: In the subgraph counting problem, we are given a (large) graph $G(V, E)$ and a (small) graph $H$ (e.g., a triangle), and the goal is to estimate the number of occurrences of $H$ in $G$. Our focus in this talk is on designing sublinear-time algorithms for approximately computing number of occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent work which primarily focused on specific families of graphs H such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$ in the literature. This is in sharp contrast to the closely related subgraph enumeration problem in which the goal is to list all copies of the subgraph $H$ in $G$. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph $H$ in a graph $G$ with $m$ edges is $O(m^{p(H)})$, where $p(H)$ is the fractional edge cover number of $H$, and enumeration algorithms with matching runtime are known for every $H$.

In this talk, we bridge this gap between the subgraph counting and subgraph enumeration problems and present a simple sublinear-time algorithm that estimates the number of occurrences of any arbitrary graph $H$ in $G$, denoted by $\#H$, to within a $(1 \pm \varepsilon)$-approximation factor with high probability in $O(m^{p(H)} /\#H)\cdot \text{poly}(\log n,1/\varepsilon)$ time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph $H$ under the additional assumption of edge-sample queries. Our results are also applicable to the more general problem of database join size estimation problem and for this slightly more general problem achieve optimal bounds for every choice of $H$.

Joint work with Michael Kapralov and Sanjeev Khanna.