# The Khot-Naor Approximation Algorithm for 3-XOR

Today I would like to discuss the Khot-Naor approximation algorithm for the 3-XOR problem, and an open question related to it.

# ARV on Abelian Cayley Graphs

Continuing from the previous post, we are going to prove the following result: let ${G}$ be a ${d}$-regular Cayley graph of an Abelian group, ${\phi(G)}$ be the normalized edge expansion of ${G}$, ${ARV(G)}$ be the value of the ARV semidefinite programming relaxation of sparsest cut on ${G}$ (we will define it below), and ${\lambda_2(G)}$ be the second smallest normalized Laplacian eigenvalue of ${G}$. Then we have

$\displaystyle \lambda_2 (G) \leq O(d) \cdot (ARV (G))^2 \ \ \ \ \ (1)$

which, together with the fact that ${ARV(G) \leq 2 \phi(G)}$ and ${\phi(G) \leq \sqrt{2 \lambda_2}}$, implies the Buser inequality

$\displaystyle \lambda_2 (G) \leq O(d) \cdot \phi^2 (G) \ \ \ \ \ (2)$

and the approximation bound

$\displaystyle \phi(G) \leq O(\sqrt d) \cdot ARV(G) \ \ \ \ \ (3)$

The proof of (1), due to Shayan Oveis Gharan and myself, is very similar to the proof by Bauer et al. of (2).

# Buser Inequalities in Graphs

As life is tentatively returning to normal, I would like to once again post technical material here. Before returning to online optimization, I would like to start with something from 2015 that we never wrote up properly, that has to do with graph curvature and with Buser inequalities in graphs.

# A Couple of Announcements

In the second week of July, 2022, there will be a summer school on algorithmic fairness at IPAM, on the UCLA campus, with Cynthia Dwork and Guy Rothblum among the lecturers. Applications (see the above link) are due by March 11, 2022.

We will soon put up a call for nominations for the test of time award to be given at FOCS 2021 (which will take place in Boulder, Colorado, in early 2022). There are three award categories, recognizing, respectively, papers from FOCS 2011, FOCS 2001, and FOCS 1991. In each category, it is also possible to nominate older papers, up to four years before the target conference. For example, for the thirty-year category, it is possible to nominate papers from FOCS 1987, FOCS 1988, FOCS 1989, FOCS 1990, in addition to the target conference FOCS 1991.

Nominations should be sent by October 31, 2021 to focs.tot.2021@gmail.com with a subject line of “FOCS Test of Time Award”. Nominations should contain an explanation of the impact of the nominated paper(s), including references to follow-on work. Self-nominations are discouraged.

In the second week of November, 2021, the Simons Institute will host a workshop on using cryptographic assumptions to prove average-case hardness of problems in high-dimensional statistics. This is such a new topic that the goal of the workshop will be more to explore new directions than to review known results, and we (think that we have) already invited all the authors of recent published work of this type. If you have proved results of this type, and you have not been invited (perhaps because your results are still unpublished?) and you would like to participate in the workshop, there is still space in the schedule so feel free to contact me or one of the other organizers. For both speakers and attendees, physical participation is preferred, but remote participation will be possible.

# The Third Annual “Why am I in Italy and you are not?” post

I moved back to Italy exactly two years ago. I was looking for some change and for new challenges and, man, talk about being careful what you wish for!

Last year was characterized by a sudden acceleration of Bocconi’s plans to develop a computer science group. From planning for a slow growth of a couple of people a year until, in 5-7 years, we could have the basis to create a new department, it was decided that a new computer science department would start operating next year — perhaps as soon as February 2022, but definitely, or at least to the extent that one can make definite plans in these crazy times, by September 2022.

Consequently, we went on a hiring spree that was surprisingly successful. Five computer scientists and four statistical physicists have accepted our offers and are coming between now and next summer. In computer science, Andrea Celli (who won the NeurIPS best paper award last year) and Marek Elias started today. Andrea, who is coming from Facebook London, works in algorithmic game theory, and Marek, who is coming TU Eindhoven, works in optimization. Within the next couple of weeks, or as soon as his visa issues are sorted out, Alon Rosen will join us from IDC Herzliya as a full professor. Readers of in theory may know Alon from his work on lattice-based cryptography, or his work on zero-knowledge, or perhaps his work on the cryptographic hardness of finding Nash equilibria. Two other computer science tenured faculty members are going to come, respectively, in February and September 2022, but I am not sure if their moves are public yet.

Meanwhile, I have been under-spending my ERC grant, but perhaps this is going to change and some of my readers will help me out.

If you are interested in coming to Milan for a post-doc, do get in touch with me. A call will be out in a month or so.

After twenty years in Northern California, I am still readjusting to seasonal weather. September is among Milan’s best months: the oppressive heat of the summer gives way to comfortable days and cool nights, but the days are still bright and sunny. Currently, there is no quarantine requirement or other travel restrictions for fully vaccinated international travellers. If you want to visit, this might be your best chance until Spring Break (last year we had a semi-lockdown from late October until after New Year, which might very well happen again; January and February are not Milan’s best months; March features spectacular cherry blossoms, and it is again an ok time to visit).

# Benny Chor

I just heard that Benny Chor died this morning. Chor did very important work on computational biology and distributed algorithms, but I (and probably many of my readers) know him primarily for his work on cryptography, for his work on randomness extraction and for introducing the notion of private information retrieval.

I only met him once, at the event for Oded Goldreich’s 60th birthday. On the occasion, he gave a talk on the Chor-Goldreich paper, which introduced the problem of randomness extraction from independent sources, and which introduced min-entropy as the right parameter by which to quantify the randomness content of random sources. He did so using the original slides used for the FOCS 1985 talk.

I took a picture during the talk, which I posted online, and later he sent me an email asking for the original. Sadly, this was the totality of our correspondence. I heard that besides being a brilliant and generous researchers, he was a very playful, likeable and nice person. My thoughts are with his family and his friends.

# The Simons Institute Reopens

This coming Fall semester the Simons Institute for the Theory of Computing in Berkeley will have in-person activities, including the really interesting program on the complexity of statistical inference, within which I will co-organize a workshop on cryptography, average-case complexity, and the complexity of statistical problems.

As it had been the case before the pandemic, all Simons Institute events will be streamed and available remotely. This includes a new series of Public Lectures called “Breakthroughs” that starts next week with a talk by Virginia Williams on matrix multiplication.

# Finally, a joy

In Rome we have an expression, mai una gioia (literally, “never (a moment of) joy”) that applies well to the present times. Yesterday, there was, finally, something to be joyous about: the announcement that two of my heroes, Laszlo Lovasz and Avi Wigderson, will share the 2021 Abel Prize, one of the highest honors of mathematics.

The reader can find a very good article about them on Quanta Magazine.

Instead of talking about their greatest accomplishment, here I would like to recall two beautiful and somewhat related results, that admit a short treatment.

# Keith Ball on Bourgain’s Legacy in Geometric Functional Analysis

The Bulletin of the AMS has just posted an article by Keith Ball on the legacy of Bourgain’s work on geometric functional analysis.

This beautifully written article talks about results and conjectures that are probably familiar to readers of in theory, but from the perspective of their mathematical motivations and of the bigger picture in which they fit.

# Post-doc Opportunities in Milan

I am recruiting two postdocs for two-year positions to work with me starting in Fall 2021 at Bocconi University. The positions have competitive salaries and are tax-free. If applicable, I will pay for relocation expenses, including the assistance of a relocation agency for help in finding a place to live and activate utilities, to complete immigration formalities, and to sign up for the national health care service.

Milan has been suffering as much or more than other European and American big cities for the effects of the Covid-19 pandemic. I have seen Milan in its normal condition for a few months from September 2019 to February 2020, and it is a beautiful cosmopolitan city, with an active cultural and social life, and with beautiful surroundings. Like San Francisco, it is smaller than one would expect it to be and very walkable (no hills!). Bocconi is situated in a semi-central area, about twenty minute walk from the Duomo.

I have received a large European grant that, besides paying for these postdoc positions, has a budget for senior visitors and for organizing two workshops over the duration of the grant. In particular, I was planning a workshop to be held last May in a villa on Lake Como. All such plans have been on hold, but Fall 2021 should be around the time that the global pandemic emergency ends, and I am planning for a lot of exciting scientific activity at Bocconi in the academic year 2021-22 and beyond.

I am looking for candidates with an established body of work on topics related to my research agenda, such as pseudorandomness and combinatorial constructions; spectral graph theory; worst-case and average-case analysis of semidefinite programming relaxation of combinatorial optimization problems.

Here are the call for applications and the application form.