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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. Faster Algorithms for High-Dimensional Robust Covariance Estimation.

   Cheng, Yu; Diakonikolas, Ilias; Ge, Rong; Woodruff, David P. (June 2019, The 32’nd Annual Conference on Learning Theory (COLT 2019))

   We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem with near-optimal error guarantees for several natural structured distributions. Our main contribution is to develop faster algorithms for this problem whose running time nearly matches that of computing the empirical covariance. Given N = Ω(d^2/\eps^2) samples from a d-dimensional Gaussian distribution, an \eps-fraction of which may be arbitrarily corrupted, our algorithm runs in time O(d^{3.26}/ poly(\eps)) and approximates the unknown covariance matrix to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees all have runtimes Ω(d^{2ω}) when \eps = Ω(1), where ω is the exponent of matrix multiplication. We also provide evidence that improving the running time of our algorithm may require new algorithmic techniques. 

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3. Computing Zigzag Persistence on Graphs in Near-Linear Time

   https://doi.org/10.4230/LIPIcs.SoCG.2021.30  

   Dey, Tamal; Hou, Tao (June 2021, Leibniz international proceedings in informatics)

   null (Ed.)

   Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general O(m^ω) time complexity are not known, where ω < 2.37286 is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with m additions and deletions on a graph with n vertices and edges, the algorithm for 0-dimension runs in O(mlog² n+mlog m) time and the algorithm for 1-dimension runs in O(mlog⁴ n) time. The algorithm for 0-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on 2-manifolds. The algorithm for 1-dimension pairs a negative edge with the earliest positive edge so that a 1-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for 0-dimension to compute the (p-1)-dimensional zigzag persistence for ℝ^p-embedded complexes in O(mlog² n+mlog m+nlog n) time. 

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4. Fast Generalized DFTs for all Finite Groups

   https://doi.org/10.1109/FOCS.2019.00052  

   Umans, Chris (November 2019, Proceedings of FOCS 2019)

   null (Ed.)

   For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^{ω/2+ε}) operations, for any ε > 0. Here, ω is the exponent of matrix multiplication. 

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5. A Faster Combinatorial Algorithm for Maximum Bipartite Matching

   Chuzhoy, Julia; Khanna, Sanjeev (January 2024, Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp [26] shows that maximum bipartite matching can be solved in O(m√n) time on a graph with n vertices and m edges. For the case of very dense graphs, a different approach based on fast matrix multiplication was subsequently developed [27, 39], that achieves a running time of O(n2.371). For the next several decades, these results represented the fastest known algorithms for the problem until in 2013, a ground-breaking work of Madry [36] gave a significantly faster algorithm for sparse graphs. Subsequently, a sequence of works developed increasingly faster algorithms for solving maximum bipartite matching, and more generally directed maximum flow, culminating in a spectacular recent breakthrough [9] that gives an m1+o(1) time algorithm for maximum bipartite matching (and more generally, for min cost flows). These more recent developments collectively represented a departure from earlier combinatorial approaches: they all utilized continuous techniques based on interior-point methods for solving linear programs. This raises a natural question: are continuous techniques essential to obtaining fast algorithms for the bipartite matching problem? Our work makes progress on this question by presenting a new, purely combinatorial algorithm for bipartite matching, that, on moderately dense graphs outperforms both Hopcroft- Karp and the fast matrix multiplication based algorithms. Similar to the classical algorithms for bipartite matching, our approach is based on iteratively augmenting a current matching using augmenting paths in the (directed) residual flow network. A common method for designing fast algorithms for directed flow problems is via the multiplicative weights update (MWU) framework, that effectively reduces the flow problem to decremental single-source shortest paths (SSSP) in directed graphs. Our main observation is that a slight modification of this reduction results in a special case of SSSP that appears significantly easier than general decremental directed SSSP. Our main technical contribution is an efficient algorithm for this special case of SSSP, that outperforms the current state of the art algorithms for general decremental SSSP with adaptive adversary, leading to a deterministic algorithm for bipartite matching, whose running time is Õ(m1/3n5/3). This new algorithm thus starts to outperform the Hopcroft-Karp algorithm in graphs with m = ω(n7/4), and it also outperforms the fast matrix multiplication based algorithms on dense graphs. We believe that this framework for obtaining faster combinatorial algorithms for bipartite matching by exploiting the special properties of the resulting decremental SSSP instances is one of the main conceptual contributions of our work that we hope paves the way for even faster combinatorial algorithms for bipartite matching. Finally, using a standard reduction from the maximum vertex-capacitated s-t flow problem in directed graphs to maximum bipartite matching, we also obtain an O(m1/3n5/3) time deterministic algorithm for maximum vertex-capacitated s-t flow when all vertex capacities are identical. 

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