Irit Dinur’s research while affiliated with Weizmann Institute of Science and other places

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness.The current workshop focused on recent developments in various sub-areas including fine-grained complexity, algorithmic fairness, pseudorandomness, cryptography, arithmetic complexity, Markov Chain Monte Carlo, structure vs. randomness in combinatorics and complexity, meta-complexity, and the complexity of approximation problems. Many of the developments are related to diverse mathematical fields such as algebra, geometry, combinatorics, analysis, and coding theory.

We give a structure theorem for Boolean functions on the p-biased hypercube which are $\epsilon$-close to degree d in $L_2$, showing that they are close to sparse juntas. Our structure theorem implies that such functions are $O(\epsilon^{C_d} + p)$-close to constant functions. We pinpoint the exact value of the constant $C_d$. We also give an analogous result for monotone Boolean functions on the biased hypercube which are $\epsilon$-close to degree d in $L_2$, showing that they are close to sparse DNFs. Our structure theorems are optimal in the following sense: for every $d,\epsilon,p$, we identify a class $\mathcal{F}_{d,\epsilon,p}$ of degree d sparse juntas which are $O(\epsilon)$-close to Boolean (in the monotone case, width d sparse DNFs) such that a Boolean function on the p-biased hypercube is $O(\epsilon)$-close to degree d in $L_2$ iff it is $O(\epsilon)$-close to a function in $\mathcal{F}_{d,\epsilon,p}$.

Property testing has been a major area of research in computer science in the last three decades. By property testing we refer to an ensemble of problems, results and algorithms which enable to deduce global information about some data by only reading small random parts of it. In recent years, this theory found its way into group theory, mainly via group stability. In this paper, we study the following problem: Devise a randomized algorithm that given a subgroup H of G, decides whether H is the whole group or a proper subgroup, by checking whether a single (random) element of G is in H. The search for such an algorithm boils down to the following purely group theoretic problem: For G of rank k, find a small as possible test subset $A\subseteq G$ such that for every proper subgroup H, $|H\cap A|\leq (1-\delta)|A|$ for some absolute constant $\delta>0$, which we call the detection probability of A. It turns out that the search for sets A of size linear in k and constant detection probability is a non-commutative analogue of the classical search for families of good error correcting codes. This paper is devoted to proving that such test subsets exist, which implies good universal error correcting codes exist -- providing a far reaching generalization of the classical result of Shannon. In addition, we study this problem in certain subclasses of groups -- such as abelian, nilpotent, and finite solvable groups -- providing different constructions of test subsets for these subclasses with various qualities. Finally, this generalized theory of non-commutative error correcting codes suggests a plethora of interesting problems and research directions.

We construct an infinite family of bounded-degree bipartite unique neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed byCapalbo et al.,our construction is simpler and may be closer to being implementable in practice, due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalises the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction, we prove a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs. Our bound generalises Kahale’s average degree bound to bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.

We describe a new parameterized family of symmetric error-correcting codes with low-density parity-check matrices (LDPC). Our codes can be described in two seemingly different ways. First, in relation to Reed-Muller codes: our codes are functions on a subset of $\mathbb{F}^n$ whose restrictions to a prescribed set of affine lines has low degree. Alternatively, they are Tanner codes on high dimensional expanders, where the coordinates of the codeword correspond to triangles of a 2-dimensional expander, such that around every edge the local view forms a Reed-Solomon codeword. For some range of parameters our codes are provably locally testable, and their dimension is some fixed power of the block length. For another range of parameters our codes have distance and dimension that are both linear in the block length, but we do not know if they are locally testable. The codes also have the multiplication property: the coordinate-wise product of two codewords is a codeword in a related code. The definition of the codes relies on the construction of a specific family of simplicial complexes which is a slight variant on the coset complexes of Kaufman and Oppenheim. We show a novel way to embed the triangles of these complexes into $\mathbb{F}^n$, with the property that links of edges embed as affine lines in $\mathbb{F}^n$. We rely on this embedding to lower bound the rate of these codes in a way that avoids constraint-counting and thereby achieves non-trivial rate even when the local codes themselves have arbitrarily small rate, and in particular below 1/2.