I attended the 62nd Midwest Theory Day last Sunday, November 13rd. Usually there are two such theory days each year, but this was the only one in 2011. The event is like an informal workshop: anyone who wishes may sign up to speak, there are also invited speakers, lunch is provided, and there is no proceedings. Nicole Immorlica organized the event, and did a great job: there was a full day of talks, the room was great, the food was as good as any conference I’ve been to, and the invited speakers were entertaining and educational.

Yael Tauman Kalai gave the first invited talk. She gave a high-level overview of some hot topics in cryptography. For example, she provided intuition into results she and co-authors obtained about designing systems whose algorithms resist *physical attacks* on cryptosystems. A dramatic physical attack is someone who wants to break into a SIM card, so cooks it in a microwave oven for a while; this can cause bits to flip in the secret key, and even modify the circuitry of the SIM card itself. (Some other examples of physical attacks are here.) Tauman Kalai presented a formalism in which the attacker could determine the system’s secret key, but would still be unable to produce a second secret key matching the system’s public key, unless the attacker used exponential resources or broke a reasonable cryptographic assumption. For a system that can defend itself against an adversary capable of obtaining continual memory leakage of the system, see the FOCS 2010 paper Cryptography Resistant to Continual Memory Leakage (co-authors Brakerski, Katz, Vaikuntanathan).

Adam Kalai was the other invited speaker; he provided an overview of his ITCS 2011 paper Compression Without a Common Prior: an Information-Theoretica Justification for Ambiguity in Natural Language (co-authors Juba, Khanna, Sudan). His talk was controversial: quite a few members of the audience either didn’t get his point, or got it and didn’t agree. The idea, as I understand it, is to produce a mechanism two computers can use to compress and decompress messages they send to each other, even if their respective compression dictionaries (or “priors”) are different. The motivation is how natural language works: you and I don’t have the same definitions for all words (and very different life experience) and yet we can communicate pretty well most of the time (for purposes of the example anyway). If the message is large enough, one could of course use an algorithm like Lempel-Ziv, which approches best-possible compression in the limit. But perhaps we just want to send one image, instead of thousands, and we would like to take advantage of the fact that both computers have useful priors for what an image is, even if those priors are not the same. The paper provides an information-theoretic framework for this, but there is, as yet, no implementable algorithm.

There was also a day full of 15- and 20-minute talks by students, postdocs and faculty. I will completely arbitrarily limit myself to the talks of the two people who sat on either side of me at lunch. Fortunately, I thought both their talks were great.

Paolo Codenotti presented some brand new work on the group isomorphism problem: Polynomial-Time Isomorphism Test for Groups with no Abelian Normal Subgroups (co-authors Babai, Qiao). This work extends the SODA 2011 paper Code Equivalence and Group Isomorphism (co-authors Babai, Grochow, Qiao). The problem solved is the following: let $latex G$ be a group on $latex n$ elements, given to us as its multiplication table of size $latex n^2$. Let $latex H$ be another group on $latex n$ elements, given in the same way. If we know that $latex G$ and $latex H$ contain no normal subgroups that are abelian (i.e., commutative), then we can determine whether $latex G$ and $latex H$ are isomorphic in polynomial time. One example of such a group is $latex A_5$, the alternating group on 5 elements, which has order $latex n=60$. Another example would be any group built from direct sums of copies of $latex A_5$.

Tyson Williams gave an overview of his ITCS 2011 paper, Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy (co-authors Cai, Kowalczyk). Williams has also posted his slides from this talk online. This paper is in the general category of holographic algorithms, a remarkable subfield of counting complexity started by Les Valiant and explored by Jin-Yai Cai, Williams’s advisor. Since the slides are online, and the intuition behind the paper’s approach is so visual, I will let the slides speak for themselves, and close this post by simply stating the paper’s main theorem.

Theorem. Over 3-regular graphs $latex G$, the counting problem for any binary, complex-weighted function $latex f$, $latex \displaystyle Z(G) = \sum_{\sigma:V \rightarrow \{0,1\}} \prod_{(u,v) \in E} f(\sigma(u),\sigma(v))$ is either polynomial-time computable or #P hard. Furthermore, the complexity is efficiently decidable.

The formalism of $latex f$ captures a wide variety of problems on 3-regular graphs, such as #VertexCover (how many vertex covers are there for the input graph), which is #P hard.

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