TCS+

A carbon-free dissemination of ideas across the globe.

TCS+ talk: Wednesday, May 24 — Mohsen Ghaffari, ETH Zurich

The next TCS+ talk will take place next Wednesday, May 24th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Mohsen Ghaffari (ETH Zurich) will will speak about “On the Complexity of Local Distributed Graph 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.

We revisit the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS ’87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best known deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and potentially narrowing down this exponential gap is one of the central long-standing open questions in the area of distributed graph algorithms.

We investigate this question by introducing a complexity-theoretic framework, which allows us to shed some light on the role of randomness in the LOCAL model. In particular, we prove completeness results which, roughly speaking, show that some rather rudimentary-looking problems are “complete” for essentially all the classic local distributed problems. In other words, if any of these complete problems can be solved deterministically in $\poly\log n$ rounds in the LOCAL model, we can deterministically solve essentially all the classic LOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\poly\log n$ rounds.

Perhaps the most surprising complete problem is the following: Color the nodes of the graph red or blue such that each node of a sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized algorithm, simply color each node randomly. This completeness result can be viewed as showing that the only obstacle to getting efficient deterministic algorithms in the LOCAL model is an efficient algorithm to round fractional values into integer values, while approximately preserving some linear constraints.

This talk is based on a joint work with Fabian Kuhn and Yannic Maus.

A version of the paper can be found here: https://arxiv.org/abs/1611.02663

TCS+ talk: Wednesday, May 10 — Avishay Tal, IAS

The next TCS+ talk will take place next Wednesday, May 19th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Avishay Tal (IAS) will will speak about “Computing Requires Larger Formulas than Approximating” (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.

A de-Morgan formula over Boolean variables x_1, …, x_n is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires large formulas is a central open question in computational complexity.

In this work, we introduce a size-amplification hardness reduction for de-Morgan formulas. We show that average-case hardness implies worst-case hardness for a larger size. More precisely, if a function f cannot be computed correctly on more than 1/2 + eps of the inputs by any formula of size s, then computing f correctly on all inputs requires size ~s*log(1/eps). The tradeoff is essentially tight. Quite surprisingly, the proof relies on a result from quantum query complexity by Reichardt [SODA, 2011].

As an application, we improve the best known formula size lower bounds for explicit functions by logarithmic factors to ~n^3/log(n). In addition, we propose candidates for explicit functions that we believe have formula size ~n^4, and prove non trivial super-quadratic formula size lower bounds for them using our reduction.

TCS+ talk: Wednesday, April 26 — Santosh Vempala, Georgia Tech

The next TCS+ talk will take place next Wednesday, April 26th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Santosh Vempala (Georgia Tech) will present joint work with Yin Tat Lee to appear in STOC’17, “Sampling Polytopes: From Euclid to Riemann”  (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.

We introduce the Geodesic Walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from the interior of a convex polytope in n-dimensional Euclidean space. The walk is a simulation of a stochastic differential equation defined using a convex barrier function; each step is the solution of an ordinary differential equation. It improves the time complexity of sampling polytopes and is the first walk to achieve sub-quadratic complexity. Most of the talk will be spent introducing relevant aspects of manifolds, stochastic processes, efficient solution of differential equations, and how this walk overcomes the bottlenecks of random walks in Euclidean space. No background in string theory (or Riemannian geometry) will be assumed.

Joint work with Lee Yin Tat.

TCS+ talk: Wednesday, April 12 — Kasper Green Larsen, Aarhus

The next TCS+ talk will take place next Wednesday, April 12th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Kasper Green Larsen (Aarhus) will present joint work with Weinstein and Yu, “Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds”  (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.

This talks presents the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds.

We introduce a new method for proving dynamic cell probe lower bounds and use it to prove a $\tilde{\Omega}(\lg^{1.5} n)$ lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting over F_2 ([Pat07]). Proving an $\omega(\lg n)$ lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Patrascu’s obituary [Tho13]. This result also implies the first $\omega(\lg n)$ lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases. We derive similar lower bounds for boolean versions of dynamic polynomial evaluation and 2D rectangle stabbing, and for the (non-boolean) problems of range selection and range median.

Our technical centerpiece is a new way of “weakly” simulating dynamic
data structures using efficient one-way communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebychev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling” method of Panigrahy et al. [PTW10].

TCS+ talk: Wednesday, March 29 — Noah Stephens-Davidowitz, NYU

TCS+ is resuming! Our next talk will take place next Wednesday, March 29th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Noah Stephens-Davidowitz (NYU) will present joint work with Oded Regev on “A Reverse Minkowski Theorem”, to appear in the next STOC  (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.

A classical problem in the geometry of numbers asks us to estimate how many lattice points lie in some ball around the origin. Minkowski’s celebrated theorem gives us a tight lower bound for this number that depends only on the determinant of the lattice. (One can think of the determinant as the limit of the density of lattice points inside large balls. So, Minkowski’s theorem gives a tight lower bound on a lattice’s “local density” based on its “global density.”) We resolve a conjecture due to Dadush that gives a nearly tight converse to Minkowski’s theorem—an upper bound on the number of lattice points in a ball that depends only on the determinants of sublattices. This theorem has numerous applications, from complexity theory, to the geometry of numbers, to the behavior of Brownian motion on flat tori.

Based on joint work with Oded Regev.

TCS+ talk: Wednesday, March 1st — Josh Alman, MIT

Our next talk will take place on Wednesday, March 1st at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Josh Alman (MIT) will present joint work with Ryan Williams on “Probabilistic Rank and Matrix Rigidity”, to appear in the next STOC  (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.

A matrix M is called rigid if M has ‘high’ rank, and one needs to change ‘many’ entries of M before it has ‘low’ rank. Matrix rigidity was introduced by Leslie Valiant in 1977, as a path towards proving arithmetic circuit lower bounds. In order to prove such a bound, one needs to give an explicit rigid matrix (for certain appropriate rigidity parameters). The problem of finding such an explicit rigid matrix is still open.

One family of matrices which was conjectured to be rigid is the Walsh-Hadamard transform (a.k.a. Sylvester matrices, a.k.a. the communication matrix of Inner Product modulo 2). I will present a proof that the Walsh-Hadamard transform is, in fact, not sufficiently rigid for Valiant’s program. Our proof uses a connection from recent “polynomial method” algorithms between low rank matrices and low degree polynomials. I will also discuss a natural notion we call the probabilistic rank of a matrix, which measures the extent to which a matrix can be probabilistically approximated by low rank matrices.

Based on joint work with Ryan Williams to appear in STOC’17.

TCS+ talk: Wednesday, February 15th — Jelani Nelson, Harvard

Our next talk will take place on Wednesday, February 15th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Jelani Nelson (Harvard) will give us the final word on the J-L lemma: “Optimality of the Johnson-Lindenstrauss lemma”  (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.

We consider the question: given integers n,d > 1, and some 0 < epsilon < 1, what is the minimum value of m such that for all n-point subsets X of d-dimensional Euclidean space, there exists an embedding f from X to m-dimensional Euclidean space with distortion at most 1+epsilon? We show that for nearly the full range of interest for the parameters n, d, and epsilon, the Johnson-Lindenstrauss lemma is tight: there exists an n-point subset of d-dimensional Euclidean space such that any such f must have m in Omega(epsilon^{-2} log n).

Joint work with Kasper Green Larsen (Aarhus University).