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1. Matrix Multiplication Verification Using Coding Theory

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.42  

   Bennett, Huck; Gajulapalli, Karthik; Golovnev, Alexander; Warton, Evelyn (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kumar, Amit; Ron-Zewi, Noga (Ed.)

   We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time). To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm). Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T. We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions). 

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2. Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.55  

   Cook, Joshua; Moshkovitz, Dana (September 2024, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   We prove that for some constant a > 1, for all k ≤ a, MATIME[n^{k(1+o(1))}]/1 ⊄ SIZE[O(n^k)], for some specific o(1) function. This is a super linear polynomial circuit lower bound. Previously, Santhanam [Santhanam, 2007] showed that there exists a constant c>1 such that for all k>1: MATIME[n^{ck}]/1 ⊄ SIZE[O(n^k)]. Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c. Using ideas from Murray and Williams [Murray and Williams, 2018], one can show for all k>1: MATIME [n^{10 k²}]/1 ⊄ SIZE[O(n^k)]. To prove this result, we construct the first PCP for SPACE[n] with quasi-linear verifier time: our PCP has a Õ(n) time verifier, Õ(n) space prover, O(log(n)) queries, and polynomial alphabet size. Prior to this work, PCPs for SPACE[O(n)] had verifiers that run in Ω(n²) time. This PCP also proves that NE has MIP verifiers which run in time Õ(n). 

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3. Recursive Error Reduction for Regular Branching Programs

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

   Chattopadhyay, Eshan; Liao, Jyun-Jie (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Guruswami, Venkatesan (Ed.)

   In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020). In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs. Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length n and width w, both of which were proved in the work of Chen et al. using their error reduction: - There is a WPRG with error ε that has seed length Õ(log(n)(√{log(1/ε)}+log(w))+log(1/ε)). - There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error ±ε with space complexity Õ(log(nw)⋅log log(1/ε)). This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler. Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al. In fact, because of its simplicity, our proof of the second result directly gives a slightly stronger claim: our algorithm computes a ε-singular value approximation (a notion of approximation introduced in a recent work by Ahmadinejad, Peebles, Pyne, Sidford and Vadhan (FOCS 2023)) of the random walk matrix of the given ROBP in space Õ(log(nw)⋅log log(1/ε)). It is not clear how to get this stronger result from the previous proofs. 

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4. Low-Memory Algorithms for Online Edge Coloring

   https://doi.org/10.4230/LIPIcs.ICALP.2024.71  

   Ghosh, Prantar; Stoeckl, Manuel (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   For edge coloring, the online and the W-streaming models seem somewhat orthogonal: the former needs edges to be assigned colors immediately after insertion, typically without any space restrictions, while the latter limits memory to be sublinear in the input size but allows an edge’s color to be announced any time after its insertion. We aim for the best of both worlds by designing small-space online algorithms for edge coloring. Our online algorithms significantly improve upon the memory used by prior ones while achieving an O(1)-competitive ratio. We study the problem under both (adversarial) edge arrivals and vertex arrivals. Under vertex arrivals of any n-node graph with maximum vertex-degree Δ, our online O(Δ)-coloring algorithm uses only semi-streaming space (i.e., Õ(n) space, where the Õ(.) notation hides polylog(n) factors). Under edge arrivals, we obtain an online O(Δ)-coloring in Õ(n√Δ) space. We also achieve a smooth color-space tradeoff: for any t = O(Δ), we get an O(Δt(log²Δ))-coloring in Õ(n√{Δ/t}) space, improving upon the state of the art that used Õ(nΔ/t) space for the same number of colors. The improvements stem from extensive use of random permutations that enable us to avoid previously used colors. Most of our algorithms can be derandomized and extended to multigraphs, where edge coloring is known to be considerably harder than for simple graphs. 

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5. Constructing many faces in arrangements of lines and segments

   https://doi.org/10.20382/jocg.v14i1a11  

   Wang, Haitao (December 2023, Journal of computational geometry)

   We present new algorithms for computing many faces in arrangements of lines and segments. Given a set $$S$$ of $$n$$ lines (resp., segments) and a set $$P$$ of $$m$$ points in the plane, the problem is to compute the faces of the arrangements of $$S$$ that contain at least one point of $$P$$. For the line case, we give a deterministic algorithm of $$O(m^{2/3}n^{2/3}\log^{2/3} (n/\sqrt{m})+(m+n)\log n)$$ time. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $$\log^{2.22}n$$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $$\log^{1/3}n$$ in certain cases (e.g., when $$m=\Theta(n)$$). For the segment case, we present a deterministic algorithm of $$O(n^{2/3}m^{2/3}\log n+\tau(n\alpha^2(n)+n\log m+m)\log n)$$ time, where $$\tau=\min\{\log m,\log (n/\sqrt{m})\}$$ and $$\alpha(n)$$ is the inverse Ackermann function. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $$\log^{2.11}n$$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $$\log n$$ in certain cases (e.g., when $$m=\Theta(n)$$). We also give a randomized algorithm of $$O(m^{2/3}K^{1/3}\log n+\tau(n\alpha(n)+n\log m+m)\log n\log K)$$ expected time, where $$K$$ is the number of intersections of all segments of $$S$$. In addition, we consider the query version of the problem, that is, preprocess $$S$$ to compute the face of the arrangement of $$S$$ that contains any given query point. We present new results that improve the previous work for both the line and the segment cases. In particulary, for the line case, we build a data structure of $$O(n\log n)$$ space in $$O(n\log n)$$ randomized time, so that the face containing the query point can be obtained in $$O(\sqrt{n\log n})$$ time with high probability (more specifically, the query returns a binary search tree representing the face so that standard binary-search-based queries on the face can be handled in $$O(\log n)$$ time each and the face itself can be output explicitly in time linear in its size). 

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