Last night, two papers appeared on the quantum physics arXiv that my coauthors and I have been working on for more than a year, and that I’m pretty happy about.
The first paper, with Guy Rothblum, is Gentle Measurement of Quantum States and Differential Privacy (85 pages, to appear in STOC’2019). This is Guy’s first paper that has anything to do with quantum, and also my first paper that has anything to do with privacy. (What do I care about privacy? I just share everything on this blog…) The paper has its origin when I gave a talk at the Weizmann Institute about “shadow tomography” (a task where you have to measure quantum states very carefully to avoid destroying them), and Guy was in the audience, and he got all excited that the techniques sounded just like what they use to ensure privacy in data-mining, and I figured it was just some wacky coincidence and brushed him off, but he persisted, and it turned out that he was 100% right, and our two fields were often studying the same problems from different angles and we could prove it. Anyway, here’s the abstract:
In differential privacy (DP), we want to query a database about n users, in a way that “leaks at most ε about any individual user,” even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that “damages the states by at most α,” even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is α-gentle for small α is also O(α)-DP, and any product measurement that is ε-DP is also O(ε√n)-gentle.Illustrating the power of this connection, we apply it to the recently studied problem of shadow tomography. Given an unknown d-dimensional quantum state ρ, as well as known two-outcome measurements E_{1},…,E_{m}, shadow tomography asks us to estimate Pr[E_{i} accepts ρ], for every i∈[m], by measuring few copies of ρ. Using our connection theorem, together with a quantum analog of the so-called private multiplicative weights algorithm of Hardt and Rothblum, we give a protocol to solve this problem using O((log m)^{2}(log d)^{2}) copies of ρ, compared to Aaronson’s previous bound of ~O((log m)^{4}(log d)). Our protocol has the advantages of being online (that is, the E_{i}‘s are processed one at a time), gentle, and conceptually simple.
Other applications of our connection include new lower bounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.
The second paper, with Robin Kothari, UT Austin PhD student William Kretschmer, and Justin Thaler, is Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. Here’s the abstract:
Given only a membership oracle for S, it is well-known that approximate counting takes Θ(√(N/|S|)) quantum queries. But what if a quantum algorithm is also given “QSamples”—i.e., copies of the state |S〉=Σ_{i∈S}|i〉—or even the ability to apply reflections about |S〉? Our first main result is that, even then, the algorithm needs either Θ(√(N/|S|)) queries or else Θ(min{|S|^{1/3},√(N/|S|)}) reflections or samples. We also give matching upper bounds.We prove the lower bound using a novel generalization of the polynomial method of Beals et al. to Laurent polynomials, which can have negative exponents. We lower-bound Laurent polynomial degree using two methods: a new “explosion argument” that pits the positive- and negative-degree parts of the polynomial against each other, and a new formulation of the dual polynomials method.
Our second main result rules out the possibility of a black-box Quantum Merlin-Arthur (or QMA) protocol for proving that a set is large. More precisely, we show that, even if Arthur can make T quantum queries to the set S⊆[N], and also receives an m-qubit quantum witness from Merlin in support of S being large, we have Tm=Ω(min{|S|,√(N/|S|)}). This resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA.
Note that QMA is “stronger” than the queries+QSamples model in that Merlin’s witness can be anything, rather than just the specific state |S〉, but also “weaker” in that Merlin’s witness cannot be trusted. Intriguingly, Laurent polynomials also play a crucial role in our QMA lower bound, but in a completely different manner than in the queries+QSamples lower bound. This suggests that the “Laurent polynomial method” might be broadly useful in complexity theory.
I need to get ready for our family’s Seder now, but after that, I’m happy to answer any questions about either of these papers in the comments.
Meantime, the biggest breakthrough in quantum complexity theory of the past month isn’t either of the above: it’s the paper by Anand Natarajan and John Wright showing that MIP*, or multi-prover interactive proof systems with entangled provers, contains NEEXP, or nondeterministic doubly-exponential time (!!). I’ll try to blog about this later, but if you can’t wait, check out this excellent post by Thomas Vidick.
Authors: Scott Aaronson, Robin Kothari, William Kretschmer, Justin Thaler
Download: PDF
Abstract: This paper proves new limitations on the power of quantum computers to solve
approximate counting -- that is, multiplicatively estimating the size of a
nonempty set $S\subseteq [N]$. Given only a membership oracle for $S$, it is
well known that approximate counting takes $\Theta(\sqrt{N/|S|})$ quantum
queries. But what if a quantum algorithm is also given "QSamples"---i.e.,
copies of the state $|S\rangle = \sum_{i\in S}|i\rangle$---or even the ability
to apply reflections about $|S\rangle$? Our first main result is that, even
then, the algorithm needs either $\Theta(\sqrt{N/|S|})$ queries or else
$\Theta(\min\{|S|^{1/3},\sqrt{N/|S|}\})$ reflections or samples. We also give
matching upper bounds. We prove the lower bound using a novel generalization of
the polynomial method of Beals et al. to Laurent polynomials, which can have
negative exponents. We lower-bound Laurent polynomial degree using two methods:
a new "explosion argument" and a new formulation of the dual polynomials
method. Our second main result rules out the possibility of a black-box Quantum
Merlin-Arthur (or QMA) protocol for proving that a set is large. We show that,
even if Arthur can make $T$ quantum queries to the set $S$, and also receives
an $m$-qubit quantum witness from Merlin in support of $S$ being large, we have
$Tm=\Omega(\min\{|S|,\sqrt{N/|S|}\})$. This resolves the open problem of giving
an oracle separation between SBP and QMA. Note that QMA is "stronger" than the
queries+QSamples model in that Merlin's witness can be anything, rather than
just the specific state $|S\rangle$, but also "weaker" in that Merlin's witness
cannot be trusted. Intriguingly, Laurent polynomials also play a crucial role
in our QMA lower bound, but in a completely different manner than in the
queries+QSamples lower bound. This suggests that the "Laurent polynomial
method" might be broadly useful in complexity theory.
Authors: Penghui Yao
Download: PDF
Abstract: This paper initiates the study of a class of entangled-games, mono-state
games, denoted by $(G,\psi)$, where $G$ is a two-player one-round game and
$\psi$ is a bipartite state independent of the game $G$. In the mono-state game
$(G,\psi)$, the players are only allowed to share arbitrary copies of $\psi$.
This paper provides a doubly exponential upper bound on the copies of $\psi$
for the players to approximate the value of the game to an arbitrarily small
constant precision for any mono-state binary game $(G,\psi)$, if $\psi$ is a
noisy EPR state, which is a two-qubit state with completely mixed states as
marginals and maximal correlation less than $1$. In particular, it includes
$(1-\epsilon)|\Psi\rangle\langle\Psi|+\epsilon\frac{I_2}{2}\otimes\frac{I_2}{2}$,
an EPR state with an arbitrary depolarizing noise $\epsilon>0$. This paper
develops a series of new techniques about the Fourier analysis on matrix spaces
and proves a quantum invariance principle and a hypercontractive inequality of
random operators. The structure of the proofs is built the recent framework
about the decidability of the non-interactive simulation of joint
distributions, which is completely different from all previous
optimization-based approaches or "Tsirelson's problem"-based approaches. This
novel approach provides a new angle to study the decidability of the complexity
class MIP$^*$, a longstanding open problem in quantum complexity theory.
Authors: Mosab Hassaan, Karam Gouda
Download: PDF
Abstract: Graphs are widely used to model complicated data semantics in many
application domains. In this paper, two novel and efficient algorithms Fast-ON
and Fast-P are proposed for solving the subgraph isomorphism problem. The two
algorithms are based on Ullman algorithm [Ullmann 1976], apply vertex-at-a-time
matching manner and path-at-a-time matching manner respectively, and use
effective heuristics to cut the search space. Comparing to the well-known
algorithms, Fast-ON and Fast-P achieve up to 1-4 orders of magnitude speed-up
for both dense and sparse graph data.
Authors: Vincent Froese, Malte Renken
Download: PDF
Abstract: We study terrain visibility graphs, a well-known graph class closely related
to polygon visibility graphs in computational geometry, for which a precise
graph-theoretical characterization is still unknown. Over the last decade,
terrain visibility graphs attracted quite some attention in the context of time
series analysis with various practical applications in areas such as physics,
geography and medical sciences. We make progress in understanding terrain
visibility graphs by providing several graph-theoretic results. For example, we
show that they can contain arbitrary large holes but not large antiholes.
Moreover, we obtain two algorithmic results which are interesting from a
practical perspective. We devise a fast shortest path algorithm on arbitrary
induced subgraphs of terrain visibility graphs and a polynomial-time algorithm
for Dominating Set on special terrain visibility graphs (called funnel
visibility graphs).
Authors: Onur Çağırıcı, Subir Kumar Ghosh, Petr Hliněný, Bodhayan Roy
Download: PDF
Abstract: We study the problem of colouring the vertices of a polygon, such that every
viewer can see a unique colour. The goal is to minimize the number of colours
used. This is also known as the conflict-free chromatic guarding problem with
vertex guards (which is quite different from point guards considered in other
papers). We study the problem in two scenarios of a set of viewers. In the
first scenario, we assume that the viewers are all points of the polygon. We
solve the related problem of minimizing the number of guards and approximate
(up to only an additive error) the number of colours in the special case of
funnels. We also give an upper bound of O(log n) colours on weak-visibility
polygons which generalizes to all simple polygons. In the second scenario, we
assume that the viewers are only the vertices of the polygon. We show a lower
bound of 3 colours in the general case of simple polygons and conjecture that
this is tight. We also prove that already deciding whether 1 or 2 colours are
enough is NP-complete.
Authors: Benjamin Doerr
Download: PDF
Abstract: In the first runtime analysis of an estimation-of-distribution algorithm
(EDA) on the multi-modal jump function class, Hasen\"ohrl and Sutton (GECCO
2018) proved that the runtime of the compact genetic algorithm with suitable
parameter choice on jump functions with high probability is at most polynomial
(in the dimension) if the jump size is at most logarithmic (in the dimension),
and is at most exponential in the jump size if the jump size is
super-logarithmic. The exponential runtime guarantee was achieved with a
hypothetical population size that is also exponential in the jump size.
Consequently, this setting cannot lead to a better runtime.
In this work, we show that any choice of the hypothetical population size leads to a runtime that, with high probability, is at least exponential in the jump size. This result might be the first non-trivial exponential lower bound for EDAs that holds for arbitrary parameter settings.
And a possible approach to avoid this obstacle
Valentine Kabanets is a famous complexity theorist from Simon Fraser University. He has been at the forefront of lower bounds for over two decades.
Today we draw attention to this work and raise an idea about trying to unravel what makes circuit lower bounds hard.
He is the common author on two new papers on the Minimum Circuit Size Problem (MCSP), which belongs to but is not known to be complete or in . We posted on MCSP four years ago and mentioned his 1999 paper with Jin-Yi Cai, which gives evidence for MCSP truly being neither complete nor in . This “intermediate” status and the problem’s simplicity have raised hopes that direct attacks might succeed. The new papers prove direct lower bounds against some restricted circuit/formula models, including constant-depth circuits with mod- gates for prime. But they stop short of mod- for composite and other barrier cases.
He has a nifty research statement on his home page. It shows how derandomization, pseudorandomness, circuit complexity, and crypto combine into his two current projects. In a clickable tab for the third heading, he puts the meta-issue in pithy terms:
Why is proving circuit lower bounds so difficult?
His first answer tab speaks a connection we have also often emphasized here:
Traditionally, designing efficient algorithms is the subject of the theory of algorithms, while lower bounds are sought in complexity theory. It turns out, however, that there is a deep connection between the two directions: better algorithms (for a certain class of problems) also yield strong lower bounds (for related problems), and vice versa: strong lower bounds translate into more efficient algorithms.
Of course we agree, and we love connections shown in the new papers to problems such as distinguishing a very slightly biased coin from a true one. But we will try to supplement the algorithmic view of circuit lower bounds with a direct look at the underlying logic.
Okay we all know that circuit lower bounds are hard. For all Kabanets’ success and beautiful work—he like the rest of the complexity field—are unable to prove what we believe is true. They cannot in the full circuit model prove anything close to what is believed to be true for at least a half a century: There are explicit Boolean functions that cannot be computed by any linear size circuit.
We feel that the logical structure of lower bounds statements gives insight into their difficulty. Perhaps this is almost a tautology. Of course the logical structure of any mathematical statement helps us understand its inherent difficulty. But we believe more: That this structure can reveal quite a bit about lower bounds. Let’s take a look at lower bounds and see if this belief holds up.
In particular let’s compare the two main approaches to proving lower bounds: non-uniform and uniform. Our claim is that they have different logical structure, and that this difference explains why there is such a gap between the two. While lower bounds—non-uniform or uniform—are hard, uniform ones are at least possible now. Non-uniform lower bounds are really very difficult.
Here is one example. To prove an explicit size lower bound for Boolean circuits—we’ll be content with just a linear one—we must give a particular family of Boolean functions (each of inputs) so that:
Here is a constant and is assumed to be large enough. The terrific paper of Kazuo Iwama, Oded Lachish, Hiroki Morizumi, and Ran Raz gives explicit Boolean functions whose size for circuits with the usual not and binary and and or operators exceeds .
Let’s look at the above example more carefully. Suppose that in place of a single Boolean function on inputs we have a list of them:
Can we prove the following?
The first thing to note is the effect of letting the number of functions vary:
If , this just becomes our original explicit circuit lower bound problem.
If is a huge value, however, this becomes the exponential lower bound shown by Claude Shannon—a known quantity.
In our terms, the latter takes equal to , so that given our function list is just the list of all Boolean functions. If all we care about is an lower bound, then the high end of the range can be something like . So at the high end we have a simple counting argument for the proof but have traded away explicitness. The question will be about the tradeoffs for in-between the extremes.
The above idea that we can model the lower bound methods by controlling the length of the list of the functions is the key to our approach. Perhaps it may help to note an analogy to other famous hard problems of constructing explicit objects. In particular, let’s look at constructing transcendental numbers. Recall these are real numbers that are not algebraic: they are not roots of polynomials with integer coefficients. They include and
The Liouville numbers of Joseph Liouville.
These are explicit numbers that were proved by him in 1844 to be transcendental. In terms of our model .
The great and puzzle. This is the observation that of or , at least one is a transcendental number. In our terms this gives .
The famous theorem of Georg Cantor—read as proving the existence of transcendental numbers since algebraic ones are countable.
Here the high end of the range is as extreme as can be. Cantor’s `list’ of numbers is uncountable—in our model, is the cardinality of the real numbers. Note, the fact that his is huge, really huge, may explain why some at the time were unimpressed by this result. They wanted the ‘list’ to be small, actually they wanted . See this for a discussion of the history of these ideas.
The theorem by Waim Zudilin, in a 2001 paper, that at least one of the numbers must be irrational. It is for “irrational” not “transcendental,” but exemplified in a highly nontrivial manner. The technical point that makes this work is interactions among these numbers that cannot be captured just by considering any one of them separately. This has .
The issue is this: Suppose that we have a list of several boolean functions . Then we can join them together to form one function so that
Clearly the function is easy implies that all of the are easy. This join trick shows that we can encode several boolean functions into one function. Note, we can even make have only order where has bits.
Thus we can join any collection of functions to make a “universal” one that is at least as hard as the worst of the single functions. More precisely,
Here is the circuit complexity of the boolean function .
If is bigger than , that is if , then the joined function has more than linearly many variables. Can we possibly establish nontrivial interactions among so many functions, say ?
One can also try to get this effect with fewer or no additional variables by taking the XOR of some subset of functions in the list. If this is done randomly for each input length then one can expect hard functions to show up for many . If this process can then be de-randomized, then this may yield an explicit hard function. We wonder how this idea might meld with Andy Yao’s famous XOR Lemma and conditions to de-randomize it.
Ken and I thought about the above simple fact about joins, which seems special to functions. Joining by interleaving the decimal expansions is not an arithmetic operation. However, it appears that there may be a similar result possible for transcendental numbers.
Lemma 1 Suppose that and are real numbers. Then
is a transcendental complex number if at least one of or are transcendental.
Proof: Let be an algebraic number. Thus there must be a polynomial with integer coefficients so that
Then it follows by complex conjugation that
Therefore and are both algebraic; thus, so is their sum which is . Thus is algebraic. It follows that is also algebraic. This shows that is transcendental.
A question: Can we show that we can do a “join” operation for three or more numbers? That is given numbers can we construct a number that is transcendental if and only if at least one of is transcendental?
Is the model useful? Is it possible for it to succeed where a direct explicit argument () does not? Does it need to rise above technical dependence on the case via the join construction?
Maybe the barriers just need 3-Dan martial-arts treatment. Source—our congrats. |
The laws of physics underlying everyday life (that is excluding extreme values of energy and gravitation like black holes, dark matter and the Big Bang) are completely known.Hasn't this statement almost always been true, in the sense that the leading minds would make this claim at many times in history. The ancient Greeks probably believed they understood physics that underlies everyday life. So did physicists after Newton. Life back then not today. My everyday life involves using a GPS device that requires understanding relativistic effects and computer chips that needed other scientific advances.
Recently a Twitter account started called justsaysinmice. The only thing this account does, is to repost breathless news articles about medical research breakthroughs that fail to mention that the effect in question was only observed in mice, and then add the words “IN MICE” to them. Simple concept, but it already seems to be changing the conversation about science reporting.
It occurred to me that we could do something analogous for quantum computing. While my own deep-seated aversion to Twitter prevents me from doing it myself, which of my readers is up for starting an account that just reposts one overhyped QC article after another, while appending the words “A CLASSICAL COMPUTER COULD ALSO DO THIS” to each one?
Lancaster – Watching the outcomes of the Israeli elections (photo: Andrey Kupavskii)
I just came back from a trip to Sweden and the U.K. I was invited to Gothenburg to be the opponent for a Ph. D. Candidate Malin Palö Forsström (by now Dr. Malin Palö Forsström), who wrote her excellent Ph. D. thesis under the supervision of Jeff Steif in Chalmers University. We also used the opportunity for a lovely mini-mini-workshop
From Gothenburg I took the train to Stockholm to spend the weekend with Anders Björner and we talked about some old projects regarding algebraic shifting. We had dinner with several colleagues including Svante Linusson who is a candidate for the European parliament!
Stockholm: With Anders and Cristins in the late 80s (left, I think this was also when I was an opponent), Svante Linusson ten days ago (right)
The British Mathematical Colloquium at Lancaster was a lovely 4-day general meeting, an opportunity to meet some old and new friends (and Internet MO friend Yemon Choi in real life), and to learn about various new developments. I am aware of the fact that my list of unfulfilled promises is longer than those of most politicians, but I do hope to come back to some mathematics from this trip to Sweden and to Lancaster.
Last week’s Tuesday was election day in Israel, and as much as I like to participate (and to devote a post to election day here on the blog – in 2009, 2012, and 2015) I had to miss the election, for the first time since 1985. (I still tried to follow the outcomes in real time.)
And we are now spending a three-day vacation and doing some mild hiking in Mitzpe Ramon, in the Negev, the Israeli desert. The view around here is spectacular. I first fell in love with the sights of the Negev when I spent six months here when I was 19 (in the army). Since then we have been caming here many times over the years, and in 2002 the annual meeting of the Israeli Mathematical Union took place here, in the same hotel.
Ein Ovdat (left). The 2002 Annual meeting of the IMU (right). A large number of Israeli mathematicians come to a substantial fraction of these annual events.
One anecdote about the Israeli election is that both major political parties of Israel, the Likud, led by Benjamin (Bibi) Netanyahu that won 35 seats in the parliament and will probably lead the coalition, and the newly formed “Blue-and-White” party, led by Benny (Benjamin) Gantz that also won 35 seats and will probably lead the opposition, stand behind quantum computing!
Left – A paragraph from “Blue and White’s” charter with a pledge to quantum computing (I thank Noam Lifshitz for telling me about it). Right – a news item (click for the article) about the quantum computing vision of Netanyahu and the Likud party.
Last year I took some time off to study online convex optimization in some detail. The reason for doing that was similar to the reason why at some point I took time off to study spectral graph theory: it was coming up in several papers that I wanted to understand, and I felt that I was missing out by not mastering an important tool. In particular, I wanted to understand:
I am happy to say that, except for the “furthermore” part of (2), I achieved my goals. To digest this material a bit better, I came up with the rather ambitious plan of writing a series of posts, in which I would alternate between (i) explaining a notion or theorem from online convex optimization (at a level that someone learning about optimization or machine learning might find useful) and (ii) explaining a complexity-theoretic application. Now that a very intense Spring semester is almost over, I plan to get started on this plan, although it is not clear that I will see it through the end. So stay tuned for the forthcoming first episode, which will be about the good old multiplicative weights algorithm.
Here are in theory‘s first ever book reviews! The books are
Giorgio Garuzzo
Quando in Italia si facevano i computer
Available for free at Amazon.com and Amazon.it.
Giorgio Ausiello
The Making of a New Science
Available from Springer, as a DRM-free PDF through your academic library.
Both books talk about the early years of computing in Italy, on the industrial and academic side, respectively. They briefly intersect with the story of Olivetti’s Elea computer.
Olivetti was a company that was founded in 1908 to make typewriters, and then branched out to other office/business machines and avionics. In the 1930s, Adriano Olivetti, a son of the founder Camillo Olivetti, took over the company. Adriano Olivetti was an unusual figure of entrepreneur deeply interested in arts, humanities and social sciences, with a utopian vision of a company reinvesting its profits in its community. In the 1950s, he led the company to develop the Elea, the first Italian computer. The Elea was made with transistors, and it came out before IBM had built its own first transistor-based computer.
The development of Elea was led by Mario Tchou. Mario Tchou was a Chinese-Italian born and raised in Rome, who studied electrical engineering at the Sapienza University of Rome and then at Brooklyn Polytechnic, eventually becoming an assistant professor at Columbia University. Olivetti persuaded Tchou to move back to Italy and lead the development of Elea, whose first prototype came out in 1957.
As production was ramping up, tragedy struck: Adriano Olivetti died in 1960, and Mario Tchou died in 1961. To shore up the finances of the company, the new CEO Roberto Olivetti brought in a series of new investors, who pushed to spin off the computer business.
At that point, Olivetti was working on another revolutionary machine, the P101, a programmable desktop calculator billed as the “first desktop computer,” which came out in 1964, attracting huge interest. Nonetheless the company spun off its “computer” division into a joint venture with GE, eventually divesting of it completely. Fortunately, they kept control of the P101 project, because those working on it were careful in branding it internally as a “calculator” (not part of the of deal with GE) rather than a “computer.”
These events are narrated, with a fascinating insider view, in Garuzzo’s book.
Giorgio Ausiello is one of the founding fathers of academic computer science in Italy. His book is a professional memoir that starts in the 1960s, at the time in which he started working on his undergraduate thesis at the Istituto Nazionale per le Applicazioni del Calcolo (INAC, later renamed IAC) at the National Research Council in Rome. At that point INAC had one of Italy’s few computers, a machine bought in 1954 from the Ferranti company in Manchester (when it was installed, it was Italy’s second computer).
As narrated in a previous post, Mauro Picone, the mathematician who was leading INAC, brought Corrado Bohm to Rome to work on this computer, and Ausiello started to work with Bohm at the time in which he was just starting to think about models of computation and lambda-calculus.
Later, Ausiello visited Berkeley in the 1968-69 academic year, when Manuel Blum and Dick Karp had just joined the faculty. Ausiello took part in the first STOC, which was held in Marina del Rey in May 1969, and, later that month, he witnessed the occupation of People’s Park in Berkeley.
The Fall of 1969 marks the start of the first Italian undergraduate programs in Computer Science, in just four universities: Bari, Milan, Pisa and Torino. Back in Italy from Berkeley, Ausiello continued to work at the National Research Council in Rome.
The book continues with a behind-the-scene narration of the events that led to the founding of the EATCS professional society, the ICALP conference and the TCS journal. There is also another trip to Berkeley in the 1980s, featuring Silvio Micali and Vijay Vazirani working on their matching algorithm, and Shafi Goldwasser just arriving in Berkeley.
Methodically documented and very detail-oriented, the book is a fascinating read, although it leaves you sometimes wanting to hear more about the personalities and the stories of the people involved and less about the attendance lists of certain meetings.
Even when it comes to the dryer details, however, I am happy that the books documents them and makes them available to future generations that will not have any living memory of the 1960s and 1970s.
I should also mention that Alon Rosen has recently interviewed Christos Papadimitriou and Avi Wigderson and those (long) interviews are full of good stories. Finally, the Simons Foundation site has an interview of Laszlo Lovasz in conversation with Avi Wigderson which I very highly recommend everybody to watch.
“You know how the \hat command in LaTeΧ puts a caret above a letter? … Well I was thinking it would be funny if someone made a package that made the \hat command put a picture of an actual hat on the symbol instead?” And then Matthew Scroggs and Adam Townsend went ahead and did it ().
Generating 4d polyhedra from their symmetries (, via), by Mikael Hvidtfeldt Christensen.
Windows on Theory and Scott Aaronson both warn about a fake web site for FOCS 2019, whose submission deadline just passed ().
GeometrieWerkstatt Gallery (). A collection of weirdly-shaped mathematical surfaces, mostly of constant mean curvature.
Sandi Irani wins IEEE TCMF Distinguished Service Award (). The award recognizes her work chairing the ad hoc committee to combat harassment and discrimination in the theory of computing community, and then getting many theory conferences to follow its recommendations.
Periodic billiard paths (). If the boundary of a given polygon is made of mirrors, these are paths that a laser beam could take that would eventually reflect back to the starting point and angle and then repeat infinitely. It remains a heavily-studied open question whether such paths exist in every triangle. This blog post from 2014 provides a proof that they do exist in polygons whose vertex angles are all rational multiples of .
PLOS disappears one (or maybe more) of its hosted blogs () without any warning to the blog author, without any attempt at keeping old blog links still working, and with only a belated apology.
The Supreme Court’s math problem (, via). Jordan Ellenberg explains why, in testing for gerrymandering, asking about deviation from proportional representation is the wrong question. Democratic systems naturally concentrate power to the majority rather than being proportional. The right question is whether that concentration is at the natural level, or is artificially accelerated in one direction or another.
EU falsely calls Internet Archive’s major collection pages, scholarly articles, and copies of US government publications “terrorism” and demands they be taken down from the internet (, see also). The EU is about to vote to require terrorism takedowns to happen within an hour, and these requests are coming on European times when all Internet Archive employees (in California) are asleep, making manual review of these bad takedowns difficult.
SIAM-ACM Conference on Algorithmic Principles of Computer Systems, APOCS (). This is a new conference to be held with SODA, next January in Salt Lake City, covering “all areas of algorithms and architectures that offer insight into the performance and design of computer systems”. Submission titles and abstracts are due August 9 (with full papers due a week later) so if this is an area you’re interested in there’s still plenty of time to come up with something to submit.
Sad news from Berkeley: Elwyn Berlekamp has died (, see also). Berlekamp made significant contributions to combinatorial game theory (motivated, as I understand it, by the mathematical study of Go endgames), coding theory, and algorithms for polynomials.
An unofficially-proposed new Berlin transit map replaces stylized axis-parallel and diagonal line segments with smooth curves (, via). The old design was seen as “out of style”, “too robotized”, and too difficult to follow routes. There’s still a strong preference for axis-parallel and diagonal lines in the new map, but the connections between them have been smoothed out.
Disclaimer: This blog post is not intended to offend University of Pennsylvania or IIT Kharagpur or Yahoo or any individuals or parties involved. The goal of this blog post is to point out some of the inefficiencies and acknowledge them as one of the motivations behind developing TrueCerts platform.
The year was 2005. I was applying for a PhD program in Computer Science in the top US universities. I have applied to 14 US universities. On Dec 2nd 2015, I received the following email (see the screenshot below) from the University of Pennsylvania (UPenn), Penn Engineering, Office of Academic Programs, Graduate Admissions. I have redacted the name, email address and the phone numbers in the emails, to preserve their privacy.
Here are the photos of the original transcript (I received from IIT Kharagpur when I graduated) and the additional transcripts (I received from IIT Kharagpur when I requested them for my PhD application). They are all laminated by IIT Kharagpur and sent to me. Feel free to laugh at my appearance on the transcript. I do too
IIT Kharagpur stated the following in their transcripts policy.
“Institute does not take responsibility of sending the duplicate copy of grade cards (transcripts) directly to other institutions/organizations, in connection with the applicants’ admission/employment etc.”
My reply to the above email and the response from UPenn are shown in the following screenshot.
In Summary, UPenn was concerned that my transcript is laminated and opened. Surprised at their response (“Your application will not go any further with the opened transcript”), I have spent couple of hours searching online and discovered that there is a lot of scam involving fake degrees and fake transcripts. There are “professionals” in India and China creating the “highest quality fakes”. These fakes are often laminated. So, UPenn decided not to process any applications with opened and laminated credentials. All my credentials are valid and correct, but my application was not processed because IIT Kharagpur has no way to “securely send” valid transcripts and UPenn has no way to “efficiently validate applicants’ transcripts”.
I have applied to 14 universities and some of them rejected me because ‘they found a better candidate’ or ‘the research group is not looking for new PhD students’ etc etc. But my UPenn application not going further because of an “inefficient and broken system” is frustrating, to say the least.
After a week, I have recovered from this frustration and went back to my daily routine of reading research papers on Theoretical Computer Science. I said to myself “I will get into one of the remaining 13 universities and I have to focus on research and resolve the P vs NP problem”
On a lighter note: There is a simple way to verify that my transcript is valid — ‘Simply open it and look at my grades’. I have received several B’s, some C’s and even couple of D’s. My CGPA is just average. Nobody in their right senses would create a fake transcript with those grades During the last decade, whenever I met any IITian I first set them up telling my credentials (B-Tech from IIT Kharagpur, Masters from USC, PhD from GeorgiaTech, Taught at Princeton) and after they say “wow”, I bet with them that their CGPA at IIT is greater than mine. I never lost till date. I take pride in this fact In my defense, I was an all-rounder at IIT Kharagpur, balancing studies, serving as a ‘secretary of fine arts, modeling and dramatics’ of my hostel, painting, learning guitar and many more things.
A second incident: During summer 2010 (at the end of my fourth year as a PhD student at GeorgiaTech), I was offered an internship at Yahoo labs and the Yahoo HR team said they want to verify all my credentials (my IIT B-Tech degree, USC Master’s degree, work experience). Yahoo uses a third-party service to rigorously verify all credentials of potential employees / interns. Yahoo fired their own CEO when they found that he lied in his resume. This process is very rigorous, but very time-consuming. The verification of all my credentials took more than couple of months. Meanwhile, I was waiting with my fingers-crossed (figuratively speaking) and hoping these verifications happen soon, so that I can start my internship and earn some serious summer money for two months and see a bank balance of more than $1,000 dollars for the first time during my grad school
Later that year, when I discovered Bitcoin white paper and the underlying Blockchain, my first thought was to use the technology to build a ‘document integrity platform’ to issue tamper-proof credentials (degrees, transcripts, employment certificates, Visas, Identity documents, driver’s license etc), which can be verified instantaneously and securely on the blockchain.
Early 2011, I got busy with writing thesis and joined the CS department Princeton University in September 2011. I have spent the first couple of years teaching at Princeton, mining Bitcoin, keeping track of Blockchain news and hoping that somebody will develop a ‘document integrity platform’. To my relief, some entrepreneurs tried to develop a ‘credential verification solutions’. To my frustration, none of those solutions are perfect. So I started preparing myself to become a full-time entrepreneur and left Princeton in summer 2015.
Today, I am very glad we have a complete data integrity solution (for universities, employers and enterprises) and I am very excited that we are preventing fraud and corruption in several areas using our TrueCerts platform.
Every week I read several stories about fraud and corruption online (Eg: the recent college admissions scandal). I approach the involved parties and explain how such instances can be rigorously prevented using technology.
I feel very fortunate to have discovered an exciting vision towards a fraud-free and efficient future. This discovery happened through the above mentioned unfortunate events. Sometimes the lowest points in your life have the greatest potential to show you the right path to your highest points.
by Abby van Soest and Elad Hazan, based on this paper
When humans interact with the environment, we receive a series of signals that indicate the value of our actions. We know that chocolate tastes good, sunburns feel bad, and certain actions generate praise or disapproval from others. Generally speaking, we learn from these signals and adapt our behavior in order to get more positive “rewards” and fewer negative ones.
Reinforcement learning (RL) is a sub-field of machine learning that formally models this setting of learning through interaction in a reactive environment. In RL, we have an agent and an environment. The agent observes its position (or “state”) in the environment and takes actions that transition it to a new state. The environment looks at an agent’s state and hands out rewards based on a hidden set of criteria.
Source: Reinforcement Learning: An Introduction. Sutton & Barto, 2017.
Typically, the goal of RL is for the agent to learn behavior that maximizes the total reward it receives from the environment. This methodology has led to some notable successes: machines have learned how to play Atari games, how to beat human masters of Go, and how to write long-form responses to an essay prompt.
It seems like a paradoxical question to ask, given that RL is all about rewards. But even though the reward paradigm is fundamentally flexible in many ways, it is also brittle and limits the agent’s ability to learn about its environment. This is due to several reasons. First, a reward signal directs the agents towards a single specific goal that may not generalize. Second, the reward signal may be sparse and uninformative, as we illustrate below.
Imagine that you want a robot to learn to navigate through the following maze.
Case 1: Sparse Rewards. The agent gets a reward of +1 when it exits the maze, and a reward of 0 everywhere else. The agent doesn’t learn anything until it stumbles upon the exit.
⇒ Clear reward signals are not always available.
Case 2: Misleading Rewards. The agent gets a reward of +1 at the entrance and a reward of +10 at the exit. The agent incorrectly learns to sit at the entrance because it hasn’t explored its environment sufficiently.
⇒ Rewards can prevent discovery of the full environment.
These issues are easy to overcome in the small maze on the left. But what about the maze on the right? As the size of the environment grows, it’ll get harder and harder to find the correct solution — the intractability of the problem scales exponentially.
So what we find is that there is power in being able to learn effectively in the absence of rewards. This intuition is supported by a body of research that shows learning fails when rewards aren’t dense or are poorly shaped; and fixing these problems can require substantial engineering effort.
By enabling agents to discover the environment without the requirement of a reward signal, we create a more flexible and generalizable form of reinforcement learning. This framework can be considered a form of “unsupervised” RL. Rather than relying on explicit and inherently limited signals (or “labels”), we can deal with a broad, unlabelled pool of data. Learning from this pool facilitates a more general extraction of knowledge from the environment.
In recent work, we propose finding a policy that maximizes entropy (which we refer to as a MaxEnt policy), or another related and concave function of the distribution. This objective is reward-independent and favors exploration.
In the video below, a two-dimensional cheetah robot learns to run backwards and forwards, move its legs fast and in all different directions, and even do flips. The cheetah doesn’t have access to any external rewards; it only uses signals from the MaxEnt policy.
Entropy is a function of the distribution over states. A high entropy distribution visits all states with near-equal frequency — it’s a uniform distribution. On the other hand, a low entropy distribution is biased toward visiting some states more frequently than others. (In the maze example, a low entropy distribution would result from the agent sitting at the entrance of the maze forever.)
So given that policy creates a distribution over states, the problem we are hoping to solve is:
When we know all the states, actions, and dynamics of a given environment, finding the policy with maximum entropy is a concave optimization problem. This type of problem can be easily and exactly solved by convex programming.
But we very rarely have all that knowledge available to use. In practice, one of several complications usually arise:
In such cases, the problem of finding a max-entropy policy becomes non-convex and computationally hard.
So what to do? If we look at many practical RL problems (Atari, OpenAI Gym), we see that there are many known, efficient solvers that can construct an optimal (or nearly-optimal) policy when they are given a reward signal.
We thus consider an oracle model: let’s assume that we have access to one of these solvers, so that we can pass it an explicit reward vector and receive an optimal policy in return. Can we now maximize entropy in a provably efficient way? In other words, is it possible to reduce this high complexity optimization problem to that of “standard” RL?
Our approach does exactly that! It is based on the Frank-Wolfe method. This is a gradient-based optimization algorithm that is particularly suited for oracle-based optimization. Instead of moving in the direction of the steepest decline of the objective function, the Frank-Wolfe method iteratively moves in the direction of the optimal point in the direction of the gradient. This is depicted below (and deserves a separate post…). The Frank-Wolfe method is a projection-free algorithm, see this exposition about its theoretical properties.
For the exact specification of the algorithm and its performance guarantee, see our paper.
To complement the theory, we also created some experiments to test the MaxEnt algorithm on simulated robotic locomotion tasks (open source code available here). We used test environments from OpenAI Gym and Mujoco and trained MaxEnt experts for various environments.
These are some results from the Humanoid experiment, where the agent is a human-like bipedal robot. The behavior of the MaxEnt agent (blue) is baselined against a random agent (orange), who explores by sampling randomly from the environment. This random approach is often used in practice for epsilon-greedy RL exploration.
In this figure, we see that over the course of 25 epochs, the MaxEnt agent progressively increases the total entropy over the state space.
Here, we see a visualization of the Humanoid’s coverage of the $xy$-plane, where the shown plane is of size 40-by-40. After one epoch, there is minimal coverage of the area. But by the 5th, 10th, and 15th epoch, we see that the agent has learned to visit all the different states in the plane, obtaining full and nearly uniform coverage of the grid!
Color the points of an grid with two colors. How big a monochromatic grid-like subset can you find? By “grid-like” I mean that it should be possible to place equally many horizontal and vertical lines, partitioning the plane into cells each of which contains a single point.
So for the coloring of the grid below, there are several monochromatic grid-like subsets. The image below shows one, and the completely red and blue southwest and northeast quadrants provide two others. The blue quadrant prevents any red grid-like subset from being larger than , and vice versa, so these are the largest grid-like subsets in this grid.
It’s not hard to prove that there always exists a monochromatic grid-like subset of size at least . Just use vertical and horizontal lines to partition the big grid into blocks of that size. If one of those blocks is monochromatic, then it’s the grid-like subset you’re looking for. And if not, you can choose a red point from each block to get a grid-like subset of the same size.
In the other direction, there exist colorings of an grid for which the largest monochromatic grid-like subset has size only a little bigger, . To find such a coloring, partition the big grid into square blocks of size , and make each block monochromatic with a randomly chosen color.
Now, consider any partition by axis-parallel lines into (irregular) rectangles, each containing one of the points of a grid-like subset. Only one row or column of the rectangles can cross each line of the partition into square blocks, so the number of rectangles that include parts of two or more square blocks is . Any remaining rectangles of the partition must come from a grid-like subset of square blocks that are all colored the same as each other. But with a random coloring, the expected size of this largest monochromatic subset of square blocks is . Therefore, the number of rectangles that stay within a single square block is limited to the total number of points in this grid-like subset of square blocks, which is again .
I’m not sure how to eliminate the remaining gap between these two bounds, or which way it should go.
One application of these ideas involves the theory of superpatterns, permutations that contain as a pattern every smaller permutation up to some size . If is a superpattern for the permutations of size , then we can obtain a point set by interpreting the position and value of each element of as Cartesian coordinates. This point set includes a grid-like subset of size , coming from a permutation of size that translates to a grid-like set of points. If the elements of the superpattern are colored with two colors, there still exists a monochromatic grid-like subset of size . And this monochromatic grid-like subset corresponds to a superpattern, for permutations of size . So, whenever the elements of a superpattern are colored with two (or finitely many) colors, there remains a monochromatic subset of elements that is still a superpattern for permutations of some smaller but non-constant size.
The study of entanglement through the length of interactive proof systems has been one of the most productive applications of complexity theory to the physical sciences that I know of. Last week Anand Natarajan and John Wright, postdoctoral scholars at Caltech and MIT respectively, added a major stone to this line of work. Anand & John (hereafter “NW”) establish the following wild claim: it is possible for a classical polynomial-time verifier to decide membership in any language in non-deterministic doubly exponential time by asking questions to two infinitely powerful, but untrusted, provers sharing entanglement. In symbols, NEEXP MIP! (The last symbol is for emphasis — no, we don’t have an MIP! class — yet.)
What is amazing about this result is the formidable gap between the complexity of the verifier and the complexity of the language being verified. We know since the 90s that the use of interaction and randomness can greatly expand the power of polynomial-time verifiers, from NP to PSPACE (with a single prover) and NEXP (with two provers). As a result of the work of Natarajan and Wright, we now know that yet an additional ingredient, the use of entanglement between the provers, can be leveraged by the verifier — the same verifier as in the previous results, a classical randomized polynomial-time machine — to obtain an exponential increase in its verification power. Randomness and interaction brought us one exponential; entanglement gives us another.
To gain intuition for the result consider first the structure of a classical two-prover one-round interactive proof system for non-deterministic doubly exponential time, with exponential-time verifier. Cutting some corners, such a protocol can be obtained by “scaling up” a standard two-prover protocol for non-deterministic singly exponential time. In the protocol, the verifier would sample a pair of exponential-length questions , send and to each prover, receive answers and , and perform an exponential-time computation that verifies some predicate about .
How can entanglement help design an exponentially more efficient protocol? At first it may seem like a polynomial-time verifier has no way to even get started: if it can only communicate polynomial-length messages with the provers, how can it leverage their power? And indeed, if the provers are classical, it can’t: it is known that even with a polynomial number of provers, and polynomially many rounds of interaction, a polynomial-time verifier cannot decide any language beyond NEXP.
But the provers in the NW protocol are not classical. They can share entanglement. How can the verifier exploit this to its advantage? The key property that is needed is know as the rigidity of entanglement. In words, rigidity is the idea that by verifying the presence of certain statistical correlations between the provers’ questions and answers the verifier can determine precisely (up to a local change of basis) the quantum state and measurements that the provers must have been using to generate their answers. The most famous example of rigidity is the CHSH game: as already shown by Werner and Summers in 1982, the CHSH game can only be optimally, or even near-optimally, won by measuring a maximally entangled state using two mutually unbiased bases for each player. No other state or measurements will do, unless they trivially imply an EPR pair and mutually unbiased bases (such as a state that is the tensor product of an EPR pair with an additional entangled state).
Rigidity gives the verifier control over the provers’ use of their entanglement. The simplest use of this is for the verifier to force the provers to share a certain number of EPR pairs and measure them to obtain identical uniformly distributed -bit strings. Such a test for EPR pairs can be constructed from CHSH games. In a paper with Natarajan we give a more efficient test that only requires questions and answers of length that is poly-logarithmic in . Interestingly, the test is built on classical machinery — the low-degree test — that plays a central role in the analysis of some classical multi-prover proof systems for NEXP.
At this point we have made an inch of progress: it is possible for a polynomial-time (in ) verifier to “command” two quantum provers sharing entanglement to share EPR pairs, and measure them in identical bases to obtain identical uniformly random -bit strings. What is this useful for? Not much — yet. But here comes the main insight in NW: suppose we could similarly force the provers to generate, not identical uniformly random strings, but a pair of -bit strings that is distributed as a pair of questions from the verifier in the aforementioned interactive proof system for NEEXP with exponential-time (in ) verifier. Then we could use a polynomial-time (in ) verifier to “command” the provers to generate their exponentially-long questions by themselves. The provers would then compute answers as in the NEEXP protocol. Finally, they would prove to the verifier, using a polynomial interaction, that is a valid pair of answers to the pair of questions — indeed, the latter verification is an NEXP problem, hence can be verified using a protocol with polynomial-time verifier.
Sounds crazy? Yes. But they did it! Of course there are many issues with the brief summary above — for example, how does the verifier even know the questions sampled by the provers? The answer is that it doesn’t need to know the entire question; only that it was sampled correctly, and that the quadruple satisfies the verification predicate of the exponential-time verifier. This can be verified using a polynomial-time interactive proof.
Diving in, the most interesting insight in the NW construction is what they call “introspection”. What makes multi-prover proof systems powerful is the ability for the verifier to send correlated questions to the provers, in a way such that each prover has only partial information about the other’s question — informally, the verifier plays a variant of prisonner’s dilemma with the provers. In particular, any interesting distribution will have the property that and are not fully correlated. For a concrete example think of the “planes-vs-lines” distribution, where is a uniformly random plane and a uniformly random line in . The aforementioned test for EPR pairs can be used to force both provers to sample the same uniformly random plane . But how does the verifier ensure that one of the provers “forgets” parts of the plane, to only remember a uniformly random line that is contained in it? NW’s insight is that the information present in a quantum state — such as the prover’s half-EPR pairs — can be “erased” by commanding the prover to perform a measurement in the wrong basis — a basis that is mutually unbiased with the basis used by the other prover to obtain its share of the query. Building on this idea, NW develop a battery of delicate tests that provide the verifier the ability to control precisely what information gets distributed to each prover. This allows a polynomial-time verifier to perfectly simulate the local environment that the exponential-time verifier would have created for the provers in a protocol for NEEXP, thus simulating the latter protocol with exponentially less resources.
One of the aspects of the NW result I like best is that they showed how the “history state barrier” could be overcome. Previous works attempting to establish strong lower bounds on the class MIP, such as the paper by Yuen et al., relies on a compression technique that requires the provers to share a history state of the computation performed by a larger protocol. Unfortunately, history states are very non-robust, and as a result such works only succeeded in developing protocols with vanishing completeness-soundness gap. NW entirely bypass the use of history states, and this allows them to maintain a constant gap.
Seven years ago Tsuyoshi Ito and I showed that MIP contains NEXP. At the time, we thought this may be the end of the story — although it seemed challenging, surely someone would eventually prove a matching upper bound. Natarajan and Wright have defeated this expectation by showing that MIP contains NEEXP. What next? NEEEXP? The halting problem? I hope to make this the topic of a future post.
The Department of Informatics at the University of Bergen (Norway) has announced a 3 years PhD position in Algorithms. The focus of the position will be on the algorithmic aspects of automated reasoning, and in particular, in the algorithms aspects of automated theorem proving.
Website: https://www.jobbnorge.no/en/available-jobs/job/168659/phd-position-in-algorithms
Email: mateus.oliveira@uib.no
I was saddened about the results of the Israeli election. The Beresheet lander, which lost its engine and crashed onto the moon as I was writing this, seems like a fitting symbol for where the country is now headed. Whatever he was at the start of his career, Netanyahu has become the Israeli Trump—a breathtakingly corrupt strongman who appeals to voters’ basest impulses, sacrifices his country’s future and standing in the world for short-term electoral gain, considers himself unbound by laws, recklessly incites hatred of minorities, and (despite zero personal piety) cynically panders to religious fundamentalists who help keep him in power. Just like with Trump, it’s probably futile to hope that lawyers will swoop in and free the nation from the demagogue’s grip: legal systems simply aren’t designed for assaults of this magnitude.
(If, for example, you were designing the US Constitution, how would you guard against a presidential candidate who openly supported and was supported by a hostile foreign power, and won anyway? Would it even occur to you to include such possibilities in your definitions of concepts like “treason” or “collusion”?)
The original Zionist project—the secular, democratic vision of Herzl and Ben-Gurion and Weizmann and Einstein, which the Nazis turned from a dream to a necessity—matters more to me than most things in this world, and that was true even before I’d spent time in Israel and before I had a wife and kids who are Israeli citizens. It would be depressing if, after a century of wildly improbable victories against external threats, Herzl’s project were finally to eat itself from the inside. Of course I have analogous worries (scaled up by two orders of magnitude) for the US—not to mention the UK, India, Brazil, Poland, Hungary, the Philippines … the world is now engaged in a massive test of whether Enlightenment liberalism can long endure, or whether it’s just a metastable state between one Dark Age and the next. (And to think that people have accused me of uncritical agreement with Steven Pinker, the world’s foremost optimist!)
In times like this, one takes one’s happiness where one can find it.
So, yeah: I’m happy that there’s now an “image of a black hole” (or rather, of radio waves being bent around a black hole’s silhouette). If you really want to understand what the now-ubiquitous image is showing, I strongly recommend this guide by Matt Strassler.
I’m happy that integer multiplication can apparently now be done in O(n log n), capping a decades-long quest (see also here).
I’m happy that there’s now apparently a spectacular fossil record of the first minutes after the asteroid impact that ended the Cretaceous period. Even better will be if this finally proves that, yes, some non-avian dinosaurs were still alive on impact day, and had not gone extinct due to unrelated climate changes slightly earlier. (Last week, my 6-year-old daughter sang a song in a school play about how “no one knows what killed the dinosaurs”—the verses ran through the asteroid and several other possibilities. I wonder if they’ll retire that song next year.) If you haven’t yet read it, the New Yorker piece on this is a must.
I’m happy that my good friend Zach Weinersmith (legendary author of SMBC Comics), as well as the GMU economist Bryan Caplan (rabblerousing author of The Case Against Education, which I reviewed here), have a new book out: a graphic novel that makes a moral and practical case for open borders (!). Their book is now available for pre-order, and at least at some point cracked Amazon’s top 10. Just last week I met Bryan for the first time, when he visited UT Austin to give a talk based on the book. He said that meeting me (having known me only from the blog) was like meeting a fictional character; I said the feeling was mutual. And as for Bryan’s talk? It felt great to spend an hour visiting a fantasyland where the world’s economies are run by humane rationalist technocrats, and where walls are going down rather than up.
I’m happy that, according to this Vanity Fair article, Facebook will still ban you for writing that “men are scum” or that “women are scum”—having ultimately rejected the demands of social-justice activists that it ban only the latter sentence, not the former. According to the article, everyone on Facebook’s Community Standards committee agreed with the activists that this was the right result: dehumanizing comments about women have no place on the platform, while (superficially) dehumanizing comments about men are an important part of feminist consciousness-raising that require protection. The problem was simply that the committee couldn’t come up with any general principle that would yield that desired result, without also yielding bad results in other cases.
I’m happy that the 737 Max jets are grounded and will hopefully be fixed, with no thanks to the FAA. Strangely, while this was still the top news story, I gave a short talk about quantum computing to a group of Boeing executives who were visiting UT Austin on a previously scheduled trip. The title of the session they put me in? “Disruptive Computation.”
I’m happy that Greta Thunberg, the 15-year-old Swedish climate activist, has attracted a worldwide following and might win the Nobel Peace Prize. I hope she does—and more importantly, I hope her personal story will help galvanize the world into accepting things that it already knows are true, as with the example of Anne Frank (or for that matter, Gandhi or MLK). Confession: back when I was an adolescent, I often daydreamed about doing exactly what Thunberg is doing right now, leading a worldwide children’s climate movement. I didn’t, of course. In my defense, I wouldn’t have had the charisma for it anyway—and also, I got sidetracked by even more pressing problems, like working out the quantum query complexity of recursive Fourier sampling. But fate seems to have made an excellent choice in Thunberg. She’s not the prop of any adult—just a nerdy girl with Asperger’s who formed the idea to sit in front of Parliament every day to protest the destruction of the world, because she couldn’t understand why everyone else wasn’t.
I’m happy that the college admissions scandal has finally forced Americans to confront the farcical injustice of our current system—a system where elite colleges pretend to peer into applicants’ souls (or the souls of the essay consultants hired by the applicants’ parents?), and where they preen about the moral virtue of their “holistic, multidimensional” admissions criteria, which amount in practice to shutting out brilliant working-class Asian kids in favor of legacies and rich badminton players. Not to horn-toot, but Steven Pinker and I tried to raise the alarm about this travesty five years ago (see for example this post), and were both severely criticized for it. I do worry, though, that people will draw precisely the wrong lessons from the scandal. If, for example, a few rich parents resorted to outright cheating on the SAT—all their other forms of gaming and fraud apparently being insufficient—then the SAT itself must be to blame so we should get rid of it. In reality, the SAT (whatever its flaws) is almost the only bulwark we have against the complete collapse of college admissions offices into nightclub bouncers. This Quillette article says it much better than I can.
I’m happy that there will a symposium from May 6-9 at the University of Toronto, to honor Stephen Cook and the (approximate) 50^{th} anniversary of the discovery of NP-completeness. I’m happy that I’ll be attending and speaking there. If you’re interested, there’s still time to register!
Finally, I’m happy about the following “Sierpinskitaschen” baked by CS grad student and friend-of-the-blog Jess Sorrell, and included with her permission (Jess says she got the idea from Debs Gardner).
This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GP(p^{m}) to the problem of finding roots in GF(p) of certain other polynomials over GP(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the log of the order of the field.
Certain observations on the application of these methods to factorization of polynomials over the rational integers are also included.I think he meant what we would call polynomial time when he says algebraic.
In 1989, Berlekamp purchased the largest interest in a trading company named Axcom Trading Advisors. After the firm's futures trading algorithms were rewritten, Axcom's Medallion Fund had a return (in 1990) of 55%, net of all management fees and transaction costs. The fund has subsequently continued to realize annualized returns exceeding 30% under management by James Harris Simons and his Renaissance Technologies Corporation.
Draw a collection of quadrilaterals in the plane, meeting edge to edge, so that they don’t surround any open space (the region they cover is a topological disk) and every vertex interior to the disk touches at least four quadrilaterals. Is it always possible to color the corners of the quadrilaterals with four colors so that all four colors appear in each quadrilateral?
The graph of corners and quadrilateral edges is a squaregraph but this question is really about coloring a different and related graph, called a kinggraph, that also includes as edges the diagonals of each quad. It’s called that because one example of this kind of graph is the king’s graph describing the possible moves of a chess king on a chessboard.
The king’s graph, and kinggraphs more generally, are examples of 1-planar graphs, graphs drawn in the plane in such a way that each edge is crossed at most once. The edges of the underlying squaregraph are not crossed at all, and the diagonals of each quad only cross each other. Squaregraphs are bipartite (like every planar graph in which all faces have an even number of edges), so they can be colored with only two colors. 1-planar graphs, in general, can require six colors (for instance you can draw the complete graph as a 1-planar graph by adding diagonals to the squares of a triangular prism) and this is tight. And you can easily 4-color the king’s graph by using two colors in alternation across the even rows of the chessboard, and a different two colors across the odd rows. So the number of colors for kinggraphs should be somewhere between these two extremes, but where?
One useful and general graph coloring method is based on the degeneracy of graphs. This is the largest number such that every subgraph has a vertex with at most neighbors; one can use a greedy coloring algorithm to color any graph with colors. Kinggraphs themselves always have a vertex with at most neighbors, but unfortunately they do not have degeneracy . If you form a king’s graph on a chessboard, and then remove its four corners, you get a subgraph in which all vertices have at least four neighbors. This turns out to be as large as possible: every kinggraph has degeneracy at most four. This is because, if you consider the zones of the system of quads (strips of quads connected on opposite pairs of edges), there always exists an “outer zone”, a zone with nothing on one side of it (see the illustration, below). You can remove the vertices of the outer zone one at a time, in order from one end to the other, always removing a vertex of degree at most four, and then repeat on another outer zone until the whole graph is gone. So the degeneracy and greedy coloring method shows that you can 5-color every kinggraph, better than the 6-coloring that we get for arbitrary 1-planar graphs.
This turns out to be optimal! For a while I thought that every kinggraph must be 4-colorable, because it was true of all the small examples that I tried. But it’s not true in general, and here’s why. If you look at the zones of the 4-colored kinggraph above, you might notice a pattern. The edges that connect pairs of quads from the same zone have colorings that alternate between two different pairs of colors. For instance, we might have a zone that has red–yellow edges alternating with blue–green edges, or another zone that has red–blue edges alternating with green–yellow edges. This is true whenever a kinggraph can be 4-colored. But there are only three ways of coloring a zone (that is, of partitioning the four colors into two disjoint pairs, which alternate along the zone). And when two zones cross each other, they must be colored differently. So every 4-coloring of a kinggraph turns into a 3-coloring of its zones. But the graph that describes the zones and its crossings is a triangle-free circle graph, and vice versa: every triangle-free circle graph describes a kinggraph. And triangle-free circle graphs may sometimes need as many as five colors, in which case so does the corresponding kinggraph.
I posted an example of a squaregraph whose circle graph needs five colors on this blog in 2008. Here’s a slightly different drawing of the same graph from a later post. Because its circle graph is not 3-colorable, the corresponding kinggraph is not 4-colorable.
There are simpler examples of squaregraphs whose circle graph needs four colors. As long as the number of colors of the circle graph is more than three, the number of colors of the kinggraph will be more than four.
On the other hand, if you can color the circle graph with three colors, then it’s also possible to translate this 3-coloring of zones back into a 4-coloring of the kinggraph. Just remove an outer zone, color the remaining graph recursively, add the removed zone back, and use the color of the zone you removed to decide which colors to assign to its vertices. Unfortunately, I don’t know the computational complexity of testing whether a circle graph is 3-colorable. There was a conference paper by Walter Unger in 1992 that claimed to have a polynomial time algorithm, but without enough details and it was never published in a journal. I think we have to consider the problem of finding a coloring as still being open.
The same method also leads to an easy calculation of the number of 4-colorings (in the same sense) of the usual kind of chessboard with squares, or of a king’s graph with vertices. In this case, the zones are just the rows and columns of the chessboard. We can use one color for the rows and two for the columns, or vice versa, so the number of 3-colorings of the zones (accounting for the fact that the 2-colorings get counted twice) is . And once the coloring of the zones is chosen, the coloring of the chessboard itself is uniquely determined by the color of any of its squares, so the total number of chessboard colorings is .
ACM-SIAM Algorithmic Principles of Computer Systems (APoCS20)
https://www.siam.org/Conferences/CM/Main/apocs20January 8, 2020
Hilton Salt Lake City Center, Salt Lake City, Utah, USA
Colocated with SODA, SOSA, and Alenex
The First ACM-SIAM APoCS is sponsored by SIAM SIAG/ACDA and ACM SIGACT.
Important Dates:
August 9: Abstract Submission and Paper Registration Deadline
August 16: Full Paper Deadline
October 4: Decision Announcement
Program Chair: Bruce Maggs, Duke University and Akamai Technologies
Submissions: Contributed papers are sought in all areas of algorithms and architectures that offer insight into the performance and design of computer systems. Topics of interest include, but are not limited to algorithms and data structures for:
- Databases
- Compilers
- Emerging Architectures
- Energy Efficient Computing
- High-performance Computing
- Management of Massive Data
- Networks, including Mobile, Ad-Hoc and Sensor Networks
- Operating Systems
- Parallel and Distributed Systems
- Storage Systems
A submission must report original research that has not previously or is not concurrently being published. Manuscripts must not exceed twelve (12) single-spaced double-column pages, in addition the bibliography and any pages containing only figures. Submission must be self-contained, and any extra details may be submitted in a clearly marked appendix.
Steering Committee:
- Michael Bender
- Guy Blelloch
- Jennifer Chayes
- Martin Farach-Colton (Chair)
- Charles Leiserson
- Don Porter
- Jennifer Rexford
- Margo Seltzer
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 -node -edge graph can be disconnected by removing vertices. Although a -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 time even for and . In the simplest case where and , the bound dates five decades back to [Kleitman IEEE Trans. Circuit Theory’69]. For the general case, the best bound is [Henzinger, Rao, Gabow FOCS’96; Linial, Lovász, Wigderson FOCS’86].
In this talk, I will present a randomized Monte Carlo algorithm with time. This algorithm proves the conjecture by Aho, Hopcroft, and Ullman when up to a polylog factor, breaks the 50-year-old bound by Kleitman, is fastest for . The story is similar for the directed graphs where we obtain an algorithm running time at most .
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 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 or certify that there is no separator of size at most “near” a given seed node.
This talk is based on joint works with Danupon Nanongkai and Sorrachai Yingchareonthawornchai.
[Guest post from Virginia Vassilevska Williams –Boaz]
Barna Saha, Sofya Raskhodnikova and I are organizing the second annual TCS Women event at STOC’19. We had an event at STOC’18 and it went really well. We envision an exciting program for the TCS Women event at STOC 2019. The details about the program are forthcoming. The new TCS Women website is here: https://sigact.org/tcswomen/. There you can learn more about our initiative.
In addition to the event, we have secured generous funding (similar to last year) from the NSF and industrial partners such as Akamai, Amazon, Google and Microsoft for travel grants for women and underrepresented minority students and postdocs to attend STOC. These grants are separate from the STOC travel awards and you can apply to both.
The application deadline for the TCS Women travel scholarship is April 25th, 2019. The information on how to apply is available here https://sigact.org/tcswomen/tcs-women-travel-scholarship/. We hope to support as many women and underrepresented minority students and postdocs as possible all over the globe to come to STOC and FCRC. The participants will also have the opportunity to present their work at the STOC poster session.
If you are aware of eligible students (not only PhD) who are interested in attending STOC, please encourage them to apply.
Best wishes,
Virginia on behalf of the TCS Women organizers