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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. 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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3. Balancing Notions of Equity: Trade-offs Between Fair Portfolio Sizes and Achievable Guarantees

   https://doi.org/10.1137/1.9781611978322.33  

   Gupta, Swati; Moondra, Jai; Singh, Mohit (January 2025, Society for Industrial and Applied Mathematics)

   Motivated by fairness concerns, we study existence and computation of portfolios, defined as: given anoptimization problem with feasible solutions D, a class C of fairness objective functions, a set X of feasible solutions is an α-approximate portfolio if for each objective f in C, there is an α-approximation for f in X. We study the trade-off between the size |X| of the portfolio and its approximation factor αfor various combinatorial problems, such as scheduling, covering, and facility location, and choices of C as top-k, ordered and symmetric monotonic norms. Our results include: (i) an α-approximate portfolio of size O(log d*log(α/4))for ordered norms and lower bounds of size \Omega( log dlog α+log log d)for the problem of scheduling identical jobs on d unidentical machines, (ii) O(log n)-approximate O(log n)-sized portfolios for facility location on n points for symmetric monotonic norms, and (iii) logO(d)^(r^2)-size O(1)-approximate portfolios for ordered norms and O(log d)-approximate for symmetric monotonic norms for covering polyhedra with a constant rnumber of constraints. The latter result uses our novel OrderAndCount framework that obtains an exponentialimprovement in portfolio sizes compared to current state-of-the-art, which may be of independent interest. 

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4. Extractors for Polynomial Sources over 𝔽₂

   https://doi.org/10.4230/LIPIcs.ITCS.2024.28  

   Chattopadhyay, Eshan; Goodman, Jesse; Gurumukhani, Mohit (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Guruswami, Venkatesan (Ed.)

   We explicitly construct the first nontrivial extractors for degree d ≥ 2 polynomial sources over 𝔽₂. Our extractor requires min-entropy k ≥ n - (√{log n})/((log log n / d)^{d/2}). Previously, no constructions were known, even for min-entropy k ≥ n-1. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy k can be generated by O(k) uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below k ≥ n-o(n). In more detail, we show that sumset extractors cannot even disperse from degree 2 polynomial sources with min-entropy k ≥ n-O(n/log log n). In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction. 

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5. Log-Seed Pseudorandom Generators via Iterated Restrictions

   Doron, Dean; Hatami, Pooya; Hoza, William (July 2020, Leibniz international proceedings in informatics)

   Saraf, Shubhangi (Ed.)

   There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The "iterated restrictions" approach, pioneered by Ajtai and Wigderson [Ajtai and Wigderson, 1989], has provided PRGs with seed length polylog n or even Õ(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(-Ω̃(log n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of ∏_{i=1}^m (1 + y_i). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [Gopalan and Yehudayoff, 2014] on elementary symmetric polynomials. 

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