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1. New Pseudorandom Generators and Correlation Bounds Using Extractors

   https://doi.org/10.4230/LIPIcs.ITCS.2025.68  

   Kumar, Vinayak M (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We establish new correlation bounds and pseudorandom generators for a collection of computation models. These models are all natural generalization of structured low-degree polynomials that we did not have correlation bounds for before. In particular: - We construct a PRG for width-2 poly(n)-length branching programs which read d bits at a time with seed length 2^O(√{log n}) ⋅ d²log²(1/ε). This comes quadratically close to optimal dependence in d and log(1/ε). Improving the dependence on n would imply nontrivial PRGs for log n-degree 𝔽₂-polynomials. The previous PRG by Bogdanov, Dvir, Verbin, and Yehudayoff had an exponentially worse dependence on d with seed length of O(dlog n + d2^dlog(1/ε)). - We provide the first nontrivial (and nearly optimal) correlation bounds and PRGs against size-n^Ω(log n) AC⁰ circuits with either n^{.99} SYM gates (computing an arbitrary symmetric function) or n^{.49} THR gates (computing an arbitrary linear threshold function). This is a generalization of sparse 𝔽₂-polynomials, which can be simulated by an AC⁰ circuit with one parity gate at the top. Previous work of Servedio and Tan only handled n^{.49} SYM gates or n^{.24} THR gates, and previous work of Lovett and Srinivasan only handled polynomial-size circuits. - We give exponentially small correlation bounds against degree-n^O(1) 𝔽₂-polynomials which are set-multilinear over some arbitrary partition of the input into n^{1-O(1)} parts (noting that at n parts, we recover all low degree polynomials). This vastly generalizes correlation bounds against degree-d polynomials which are set-multilinear over a fixed partition into d blocks, which were established by Bhrushundi, Harsha, Hatami, Kopparty, and Kumar. The common technique behind all of these results is to fortify a hard function with the right type of extractor to obtain stronger correlation bounds for more general models of computation. Although this technique has been used in previous work, they rely on the model simplifying drastically under random restrictions. We view our results as a proof of concept that such fortification can be done even for classes that do not enjoy such behavior. 

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2. Sparsity Lower Bounds for Probabilistic Polynomials

   https://doi.org/10.4230/lipics.itcs.2025.3  

   Alman, Josh; Chattopadhyay, Arkadev; Williams, Ryan (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   Abstract: Probabilistic polynomials over commutative rings offer a powerful way of representing Boolean functions. Although many degree lower bounds for such representations have been proved, sparsity lower bounds (counting the number of monomials in the polynomials) have not been so common. Sparsity upper bounds are of great interest for potential algorithmic applications, since sparse probabilistic polynomials are the key technical tool behind the best known algorithms for many core problems, including dense All-Pairs Shortest Paths, and the existence of sparser polynomials would lead to breakthrough algorithms for these problems. In this paper, we prove several strong lower bounds on the sparsity of probabilistic and approximate polynomials computing Boolean functions when 0 means "false". Our main result is that the AND of n ORs of c log n variables requires probabilistic polynomials (over any commutative ring which isn't too large) of sparsity n^Ω(log c) to achieve even 1/4 error. The lower bound is tight, and it rules out a large class of polynomial-method approaches for refuting the APSP and SETH conjectures via matrix multiplication. Our other results include: - Every probabilistic polynomial (over a commutative ring) for the disjointness function on two n-bit vectors requires exponential sparsity in order to achieve exponentially low error. - A generic lower bound that any function requiring probabilistic polynomials of degree d must require probabilistic polynomials of sparsity Ω(2^d). - Building on earlier work, we consider the probabilistic rank of Boolean functions which generalizes the notion of sparsity for probabilistic polynomials, and prove separations of probabilistic rank and probabilistic sparsity. Some of our results and lemmas are basis independent. For example, over any basis {a,b} for true and false where a ≠ b, and any commutative ring R, the AND function on n variables has no probabilistic R-polynomial with 2^o(n) sparsity, o(n) degree, and 1/2^o(n) error simultaneously. This AND lower bound is our main technical lemma used in the above lower bounds. 

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3. Log-rank and lifting for AND-functions

   https://doi.org/10.1145/3406325.3450999  

   Knop, Alexander; Lovett, Shachar; McGuire, Sam; Yuan, Weiqiang (June 2021, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing)

   Let f: {0, 1}n → {0, 1} be a boolean function, and let f∧(x, y) = f(x ∧ y) denote the AND-function of f, where x ∧ y denotes bit-wise AND. We study the deterministic communication complexity of f∧ and show that, up to a logn factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of f∧. This comes within a logn factor of establishing the log-rank conjecture for AND-functions with no assumptions on f. Our result stands in contrast with previous results on special cases of the log-rank conjecture, which needed significant restrictions on f such as monotonicity or low F2-degree. Our techniques can also be used to prove (within a logn factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of f∧ is polynomially related to the AND-decision tree complexity of f. The results rely on a new structural result regarding boolean functions f: {0, 1}n → {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing f has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials f: {0, 1}n → with a larger range. 

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4. Lossy Kernelization for (Implicit) Hitting Set Problems

   https://doi.org/10.4230/LIPIcs.ESA.2023.49  

   Fomin, Fedor V; Le, Tien-Nam; Lokshtanov, Daniel; Saurabh, Saket; Thomassé, Stéphan; Zehavi, Meirav (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   We re-visit the complexity of polynomial time pre-processing (kernelization) for the d-Hitting Set problem. This is one of the most classic problems in Parameterized Complexity by itself, and, furthermore, it encompasses several other of the most well-studied problems in this field, such as Vertex Cover, Feedback Vertex Set in Tournaments (FVST) and Cluster Vertex Deletion (CVD). In fact, d-Hitting Set encompasses any deletion problem to a hereditary property that can be characterized by a finite set of forbidden induced subgraphs. With respect to bit size, the kernelization complexity of d-Hitting Set is essentially settled: there exists a kernel with 𝒪(k^d) bits (𝒪(k^d) sets and 𝒪(k^{d-1}) elements) and this it tight by the result of Dell and van Melkebeek [STOC 2010, JACM 2014]. Still, the question of whether there exists a kernel for d-Hitting Set with fewer elements has remained one of the most major open problems in Kernelization. In this paper, we first show that if we allow the kernelization to be lossy with a qualitatively better loss than the best possible approximation ratio of polynomial time approximation algorithms, then one can obtain kernels where the number of elements is linear for every fixed d. Further, based on this, we present our main result: we show that there exist approximate Turing kernelizations for d-Hitting Set that even beat the established bit-size lower bounds for exact kernelizations - in fact, we use a constant number of oracle calls, each with "near linear" (𝒪(k^{1+ε})) bit size, that is, almost the best one could hope for. Lastly, for two special cases of implicit 3-Hitting set, namely, FVST and CVD, we obtain the "best of both worlds" type of results - (1+ε)-approximate kernelizations with a linear number of vertices. In terms of size, this substantially improves the exact kernels of Fomin et al. [SODA 2018, TALG 2019], with simpler arguments. 

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5. Low-degree testing over grids

   https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.41  

   Amireddy, Prashanth; Srinivasan, Srikanth; Sudan, Madhu (January 2023, Proceedings in Informatics (LIPIcs):Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   We study the question of local testability of low (constant) degree functions from a product domain 𝒮_1 × … × 𝒮_n to a field 𝔽, where 𝒮_i ⊆ 𝔽 can be arbitrary constant sized sets. We show that this family is locally testable when the grid is "symmetric". That is, if 𝒮_i = 𝒮 for all i, there is a probabilistic algorithm using constantly many queries that distinguishes whether f has a polynomial representation of degree at most d or is Ω(1)-far from having this property. In contrast, we show that there exist asymmetric grids with |𝒮_1| = ⋯ = |𝒮_n| = 3 for which testing requires ω_n(1) queries, thereby establishing that even in the context of polynomials, local testing depends on the structure of the domain and not just the distance of the underlying code. The low-degree testing problem has been studied extensively over the years and a wide variety of tools have been applied to propose and analyze tests. Our work introduces yet another new connection in this rich field, by building low-degree tests out of tests for "junta-degrees". A function f:𝒮_1 × ⋯ × 𝒮_n → 𝒢, for an abelian group 𝒢 is said to be a junta-degree-d function if it is a sum of d-juntas. We derive our low-degree test by giving a new local test for junta-degree-d functions. For the analysis of our tests, we deduce a small-set expansion theorem for spherical/hamming noise over large grids, which may be of independent interest. 

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