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1. Local Enumeration and Majority Lower Bounds

   https://doi.org/10.4230/LIPIcs.CCC.2024.17  

   Gurumukhani, Mohit; Paturi, Ramamohan; Pudlák, Pavel; Saks, Michael; Talebanfard, Navid (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics. 

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2. Lower Envelopes of Surface Patches in 3-Space

   https://doi.org/10.4230/LIPICS.ESA.2024.6  

   Agarwal, Pankaj K; Ezra, Esther; Sharir, Micha (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   Let Σ be a collection of n surface patches, each being the graph of a partially defined semi-algebraic function of constant description complexity, and assume that any triple of them intersect in at most s = 2 points. We show that the complexity of the lower envelope of the surfaces in Σ is O(n² log^{6+ε} n), for any ε > 0. This almost settles a long-standing open problem posed by Halperin and Sharir, thirty years ago, who showed the nearly-optimal albeit weaker bound of O(n²⋅ 2^{c√{log n}}) on the complexity of the lower envelope, where c > 0 is some constant. Our approach is fairly simple and is based on hierarchical cuttings and gradations, as well as a simple charging scheme. We extend our analysis to the case s > 2, under a "favorable cross section" assumption, in which case we show that the bound on the complexity of the lower envelope is O(n² log^{11+ε} n), for any ε > 0. Incorporating these bounds with the randomized incremental construction algorithms of Boissonnat and Dobrindt, we obtain efficient constructions of lower envelopes of surface patches with the above properties, whose overall expected running time is O(n² polylog), as well as efficient data structures that support point location queries in their minimization diagrams in O(log²n) expected time. 

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3. Approximate Unitary k-Designs from Shallow, Low-Communication Circuits (Extended Abstract)

   https://doi.org/10.4230/LIPIcs.ITCS.2025.69  

   LaRacuente, Nicholas; Leditzky, Felix (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   Random unitaries are useful in quantum information and related fields but hard to generate with limited resources. An approximate unitary k-design is a measure over an ensemble of unitaries such that the average is close to a Haar (uniformly) random ensemble up to the first k moments. A strong notion of approximation bounds the distance from Haar randomness in relative error: the weighted twirl induced by an approximate design can be written as a convex combination involving that of an exact design and vice versa. The main focus of our work is on efficient constructions of approximate designs, in particular whether relative-error designs in sublinear depth are possible. We give a positive answer to this question as part of our main results: 1. Twirl-Swap-Twirl: Let A and B be systems of the same size. Consider a protocol that locally applies k-design unitaries to A^k and B^k respectively, then exchanges l qudits between each copy of A and B respectively, then again applies local k-design unitaries. This protocol yields an ε-approximate relative k-design when l = O(k log k + log(1/ε)). In particular, this bound is independent of the size of A and B as long as it is sufficiently large compared to k and 1/ε. 2. Twirl-Crosstwirl: Let A_1, … , A_P be subsystems of a multipartite system A. Consider the following protocol for k copies of A: (1) locally apply a k-design unitary to each A_p for p = 1, … , P; (2) apply a "crosstwirl" k-design unitary across a joint system combining l qudits from each A_p. Assuming each A_p’s dimension is sufficiently large compared to other parameters, one can choose l to be of the form 2 (Pk + 1) log_q k + log_q P + log_q(1/ε) + O(1) to achieve an ε-approximate relative k-design. As an intermediate step, we show that this protocol achieves a k-tensor-product-expander, in which the approximation error is in 2 → 2 norm, using communication logarithmic in k. 3. Recursive Crosstwirl: Consider an m-qudit system with connectivity given by a lattice in spatial dimension D. For every D = 1, 2, …, we give a construction of an ε-approximate relative k-design using unitaries of spatially local circuit depth O ((log m + log(1/ε) + k log k ) k polylog(k)). Moreover, across the boundaries of spatially contiguous sub-regions, unitaries used in the design ensemble require only area law communication up to corrections logarithmic in m. Hence they generate only that much entanglement on any product state input. These constructions use the alternating projection method to analyze overlapping Haar twirls, giving a bound on the convergence speed to the full twirl with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing system size. The Recursive Crosstwirl construction answers one variant of [Harrow and Mehraban, 2023, Open Problem 1], showing that with a specific, layered architecture, random circuits produce relative error k-designs in sublinear depth. Moreover, it addresses [Harrow and Mehraban, 2023, Open Problem 7], showing that structured circuits in spatial dimension D of depth << m^{1/D} may achieve approximate k-designs. 

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4. Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model

   https://doi.org/10.4230/LIPIcs.FSTTCS.2024.29  

   Konrad, Christian; McGregor, Andrew; Sengupta, Rik; Than, Cuong (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Barman, Siddharth; Lasota, Sławomir (Ed.)

   We consider the problem of estimating the size of a maximum matching in low-arboricity graphs in the dynamic graph stream model. In this setting, an algorithm with limited memory makes multiple passes over a stream of edge insertions and deletions, resulting in a low-arboricity graph. Let n be the number of vertices of the input graph, and α be its arboricity. We give the following results. 1) As our main result, we give a three-pass streaming algorithm that produces an (α + 2)(1 + ε)-approximation and uses space O(ε^{-2}⋅α²⋅n^{1/2}⋅log n). This result should be contrasted with the Ω(α^{-5/2}⋅n^{1/2}) space lower bound established by [Assadi et al., SODA'17] for one-pass algorithms, showing that, for graphs of constant arboricity, the one-pass space lower bound can be achieved in three passes (up to poly-logarithmic factors). Furthermore, we obtain a two-pass algorithm that uses space O(ε^{-2}⋅α²⋅n^{3/5}⋅log n). 2) We also give a (1+ε)-approximation multi-pass algorithm, where the space used is parameterized by an upper bound on the size of a largest matching. For example, using O(log log n) passes, the space required is O(ε^{-1}⋅α²⋅k⋅log n), where k denotes an upper bound on the size of a largest matching. Finally, we define a notion of arboricity in the context of matrices. This is a natural measure of the sparsity of a matrix that is more nuanced than simply bounding the total number of nonzero entries, but less restrictive than bounding the number of nonzero entries in each row and column. For such matrices, we exploit our results on estimating matching size to present upper bounds for the problem of rank estimation in the dynamic data stream model. 

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5. 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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