Do we have TCS “Harvey Weinsteins”?
Following the Harvey Weinstein scandal, I’d like to believe that the academic environment at large, and theoretical computer science in particular, is a much better environment than Hollywood. But we would be misleading ourselves to think that we are completely without our issues. This story can be an opportunity for each of us, men and women, to think what can we do to eradicate such behavior from our field. I don’t have any easy answers, but all of us, especially senior men in positions of influence, should try to do what we can and not look the other way. There is not much that can be more damaging for a person coming up in a field than somebody abusing and objectifying them rather than treating them as a colleague and an intellectual.
This is not exactly related, but what prompted me to write this post was hearing from a friend (who is a non CS faculty in another part of the U.S.) whose children were kicked out from a private (non religious) school when the principal learned they come from an LGBT family. I was truly shocked that something like that can happen in a fairly large U.S. city. It got me thinking (again) of how easy it is to believe that such issues are a thing of the past, or happen in only the remotest parts of the world or the country, when you are not part of the affected population.
STOC 2018 CFP
The call for paper for STOC is up, see http://acmstoc.org/stoc2018/ and here. The deadline is November 3, 2017 4:59pm Eastern Daylight Time.
STOC 2018 will again be a Theory Fest and also celebrate the 50th anniversary of STOC. As part of the “retro” atmosphere, the STOC PC also asks submissions to be sent by mail in 20 printed copies at most 10 pages.
Michael Cohen
[This is a guest blog post on behalf of MSR, communicated to me by Yuval Peres. Like everyone who knew Michael, I was incredibly shocked and saddened by the news. I had relatively few interactions with Michael, but he has made a deep impression. I was always hoping we’d get a chance to collaborate , as it was clear to me that he’s exceptionally talented and creative. If you’ve known Michael, feel free to share your perspective in the comments. Boaz]
The Microsoft Research family is shocked and deeply saddened by the tragic loss of our colleague and friend Michael Cohen. Michael was a brilliant mathematician and a rising star in his field. He spent this past summer with us as a Microsoft Research Fellow, doing what he loved most. Over the summer, he made sweeping progress in online learning and online algorithms, two fields he had just recently become acquainted with. In addition to solving five open problems in these areas, he continued his substantial progress on the kserver problem, one of the most celebrated and notoriously difficult challenges in the space of adaptive algorithms.
Michael was a truly exceptional individual. We will remember Michael for his infectious smile and his largerthanlife personality. We will never forget his unrelenting curiosity, his thirst for knowledge, and his deep love for mathematics and theoretical foundations of computing. We are stunned by his loss. We will hold onto our memories of Michael, and know that his ideas and scientific accomplishments will continue on as important advances.
We extend our most sincere condolences to Michael’s family and friends.
Special year on combinatorics and complexity
This year Harvard’s Center for Mathematical Sciences and Applications is running a special year on combinatorics and complexity. We have many longterm visitors and postdocs, and several events that theoretical computer scientists might wish to take part in, including four workshops:
 Additive Combinatorics (10/2/201710/6/2017);
 Algebraic Methods in Combinatorics (11/13/2017 11/17/2017);
 Probabilistic Methods in Combinatorics (2/5/20182/9/2018); and
 Coding and Information Theory (4/9/20184/13/2018)
Our Friday theory reading group will move to the CMSA center and feature many speakers from the special year.
In addition, the program will feature a series of public lectures by Noga Alon (Sept. 7), Jennifer Chayes (Nov. 2), Jacob Fox (Feb. 1), and Dan Spielman (March 20).
Combinatorics will also be featured in this year’s Ahlfors Lecture Series, given by Timothy Gowers (University of Cambridge) on October 1112. Leslie Valiant will give the Ding Shum lecture on October 10.
Harvard will also host the Women In Theory workshop in the summer of 2018 (details forthcoming),
Participation: Researchers interested in participating in the special year through a short or longterm visit to CMSA are encouraged to contact the organizers through the CMSA Administrative Coordinator, Sarah LaBauve (slabauve@math.harvard.edu). Each of the workshops is also open to participation by all interested researchers subject to capacity. Registration forms can be found on the webpages for the individual workshops.
FOCS 2017: Registration and call for workshops
Fall is coming, and with it the annual holiday of FOCS. FOCS 2017 will be held at Berkeley from Oct 1517, with a day of workshops/tutorials on Oct 14th. Registrations are now open at http://focs17.simons.berkeley.edu/registration.html
Early registration deadline for the conference is Sept 25th, 2017.
Speaking of workshops, there is one way to guarantee that FOCS has a workshop in an area you care about, and it is to organize such a workshop yourself!
Speaking from experience, organizing a workshop is nonzero but not unmanageable amount of work. It is a great way to connect with colleagues in your area, as well as expose some other people in the TCS community to it. The call for workshops is now up at http://focs17.simons.berkeley.edu/workshopAndTutorial.html
A proposal for a workshop can be quite minimal – the main thing you need is the names of the organizers and speakers. To submit a proposal, send an email to Aleskander Mądry and James Lee by September 3, 2017.
The intermediate complexity conjecture
One of the mysteries of computation is that, as far as we can tell, the time complexity of many natural computational problems is either (with a small exponent, such as or ) or at least (with a not too small coefficient in the exponent). This is perhaps the main reason why we are excited when people discover an (or even ) time algorithm for an important problem X: we view this as evidence that X is of the EASY type, and the exponent is likely to be improved over time (as indeed most often happens).
Similarly, this is why we care about hardness reductions, even when they hide huge constants or exponents: we view them as evidence that a problem is of the HARD type, and expect (as is also often the case) the efficiency of the reduction to be eventually improved.
This is also why our asymptotic theory of efficiency in informative in a finite world, and we don’t spend too much time worrying about the fact that for every input length that could possibly fit in the world’s storage capacity.
Of course there are results such as the time hierarchy theorem and Ladner’s theorem that tell us that “intermediate complexity” do exist, but the hope is that this doesn’t happen for “natural” problems.
(In particular, we can artificially create problems of complexity or by padding exponentially hard problems. This is one of the reasons why Impagliazzo Paturi and Zane argued that for NP problems, the parameter should be thought of the solution size and not the input size.)
One family of natural problems are constraint satisfaction problems (CSPs). For some finite alphabet and constant , a CSP is characterized by a set of predicates from .
The input for is a collection of functions , each of which applies some predicate in to a subset of its coordinates.
The basic task in CSPs is exact satisfiability, where the goal is to tell whether there is some assignment for the inputs that such that .
The celebrated CSP dichotomy conjecture (which perhaps should now be called a theorem) implies (in its modern, algebraic form) that for every , either there is a polynomialtime (in fact I believe that at most time) algorithm for the exact satisfiability of or there is a constantfactor blowup reduction from 3SAT to , implying that (under the exponentialtime hypothesis or ETH) its complexity is .
The approximation problem for is parameterized by two numbers (known as the soundness and completeness paramters) and asks to distinguish, given an input of , between the case that there is some assignment such that , and the case that for every . There is one sense in which understanding the complexity of the approximation problem is easier, since in the context of approximation, the pesky difference between 3SAT and 3XOR (bane of many failed NP vs P proofs, and the main reason the dichotomy conjecture is so hard to prove) disappears. On the other hand, in this context a problem such as BIPARTITNESS (or ), which was in the EASY column of exactly solving, become the NPhard problem of MaxCut.
One of the most fascinating consequences of the Unique Games Conjecture is that, if it is true, then there are in fact CSPs for which their approximation is neither EASY nor HARD. That is, the Unique Games Conjecture (plus the ETH) implies the following conjecture:
Intermediate Complexity Conjecture: For every , there exist some , and such that vs approximation of can be done in time but (the potentially easier) vs approximation cannot be done in time .
This is a consequence of the subexponential time algorithm for unique games, which in particular implies that for, say, , there is some that tends to zero with such that the vs problem for the unique games CSP can be solved in time .
On the other hand, the Unique Games Conjecture implies that for every , no matter how close is to zero, or is to , the vs problem for unique games is NP hard and hence (under the ETH) cannot be solved in time .
I find this conjecture very fascinating. A priori, we might expect a zero one law for CSP’s complexity, where, depending on the approximation quality, one could either solve the problem in time or require at least time. Indeed, we have good reasons to believe that predicates such as 3XOR and 3SAT satisfy such a zero/one law. But, if the intermediate complexity conjecture is true then for some predicates (such as unique games itself, and probably even max cut, which can hardly be called “unnatural”) we would get a nontrivial curve of the running time vs approximation quality, that looks something like this:
Figure 1: Conjectured time complexity of vs approximation of unique games as a function of assuming the unique games conjecture. The UGC characterizes the threshold when the time complexity becomes for some , while there is also a (not yet fully determined) threshold when there is a linear blowup reduction from 3SAT and hence (under the ETH) the complexity is .
(One other setting where people were not sure if the complexity of some finitely specified object must be either exponential or polynomial is group theory, where one can define the complexity of a finitely generated group as the number of distinct elements that can be generated by a word of length $n$. There it turned out there are groups of “intermediate complexity”.)
What is also quite interesting is that we might be able to prove the intermediate complexity conjecture without resolving the UGC. In an exciting recent work, Dinur, Khot, Kindler, Minzer and Safra gave a combinatorial hypothesis that, if true, would imply the NP hardness of vs approximation for unique games where , and can be arbitrarily close to . Thus, if their hypothesis is true then, combined with the subexponential time algorithm mentioned above, it would imply the intermediate complexity conjecture. (If the construction is improved to obtain completeness that is close to then of course it would prove the unique games conjecture as well.)
Through discussions with Pravesh Kothari and David Steurer, we showed that the DKKMS combinatorial hypothesis is equivalent to a fairly simple to state statement. Informally, it can be stated as follows: let be the graph on binary matrices, such that two matrices have an edge if has rank one over GF(2). (UGC/PCP experts might recognize this graph as the degree two short code graph.) This graph is not a small set expander in the sense that, if you think about it for few minutes, you see that there are sets matrices of measure such that if we pick a random and a random neighbor of , then with constant probability will also be in . The DKKMS hypothesis is equivalent to the conjecture that these sets that you think about after few minutes encompass all such examples.
Let me state this more formally: Lets say that a subset is a basic set if or where are column vectors. Note that each such set has measure among all matrices and that a random neighbor of an element in will be in with probability . For every constant , you can clearly construct a set where a random neighbor stays inside with probability by intersecting basic sets. You can also keep this probability constant even if you just subsampled a constant fraction of this intersection (and you can add a few random vertices as well). The DKKMS hypothesis is equivalent to the statement that this is all you can do. That is, it is equivalent to the following statement:
Inverse short code conjecture: For every there exists some and such that if is large enough and is the graph above with vertex set and if , then for every , if satisfies then there are basic sets such that satisfies .
I call this an “inverse conjecture” since it is aimed at characterizing the sets that have unusually small expansion in the shortcode graphs in terms of their correlation with basic sets.
I find it a natural question, though have to admit that, despite thinking about it some time (and also talking about it, not just with David and Pravesh but also Irit Dinur, Shachar Lovett and Kaave Hoesseini) I still don’t have a strong intuition whether it’s false or true.
Given that the short code graph is a Cayley graph over the Boolean cube, this seems like a question that could be amenable to tools from Fourier analysis and additive combinatorics.
Some partial progress has been made by the DKKMS folks in their new manuscript.
In fact, the same observations show that their original hypothesis is also equivalent not just to the “inverse short code conjecture” but also to Hypothesis 1.7 in this report, which is an analogous “inverse conjecture” for the Grassmanian graph where, for , the vertices are dimension subspaces of , and two subspaces are neighbors if their intersection has dimension .
Nevertheless, at the moment this question is still wide open, and I hope this writeup encourages more people to work on it. Other than resolving this question, there are some other interesting research directions, including:
 The DKKMS reduction combined with the Grigoriev 3XOR instance yields a concrete candidate sumofsquares integrality gap for, say, vs unique games. By construction, the SOS value of this instance is , but is analyzing the soundness of this instance easier than the general case?

The underlying building block for DKKMS is a problem known as a smooth label cover. This is not precisely a CSP, but can we show that it has intermediate complexity? i.e., give a subexponential time algorithm for it? If the DKKMS hypothesis is correct then we know that such an algorithm should be an SDP hierarchy.

Suppose the combinatorial hypothesis is true, is there an inherent reason why its proof should not be a low degree SOS proof? After all, related isoperimetric results on the short code graph have been shown by low degree SOS proofs. Also, is the proof of the current best bounds coming from the DKKMS manuscript SOSizable?

The quantiative bounds for the DKKMS reduction are terrible. I haven’t worked out all the details but it seems that the reduction from 3XOR to vs approximation of unique games would map an variable instance to an instance of at least (or maybe even ) size. Is this inherent?

Is there a general conjecture that would predict for (say even if is a single predicate) and what should be the time complexity of vs approximation for it? We have some approaches of trying to predict when the complexity should be by considering pairwise independent distributions. I feel like a conjecture trying to characterize what can be done in time would have something to do with defining a suitable notion of “$(1+\epsilon)$wise independent distributions” perhaps via concepts of mutual information. For example, perhaps we can argue that a smooth label cover is a wise independent CSP to a certain extent.