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1. Channels, Billiards, and Perfect Matching 2-Divisibility

   https://doi.org/10.37236/9151  

   Barkley, Grant T.; Liu, Ricky Ini (April 2021, The Electronic Journal of Combinatorics)

   null (Ed.)

   Let $$m_G$$ denote the number of perfect matchings of the graph $$G$$. We introduce a number of combinatorial tools for determining the parity of $$m_G$$ and giving a lower bound on the power of 2 dividing $$m_G$$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $$G$$ modulo $$2$$. A result of Lovász states that the existence of a nontrivial channel is equivalent to $$m_G$$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $$2$$ dividing $$m_G$$ when $$G$$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $$m_G$$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $$2^{\frac{\gcd(m+1,n+1)-1}{2}}$$ divides the number of domino tilings of the $$m\times n$$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid. 

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2. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza; Galkowski, Jeffrey (June 2023, Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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3. Set Cover in Sub-linear Time

   Indyk, Piotr; Mahabadi, Sepideh; Rubinfeld, Ronitt; Vakilian, Ali; Yodpinyanee, Anak (January 2018, Annual ACM-SIAM Symposium on Discrete Algorithms)

   We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)^1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is , even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)-approximation algorithm with Õ(mn/kε^2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of query complexity. Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size αk, with the key property that a small number of “almost uniformly distributed” modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems. 

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4. Zero Cycles on a Product of Elliptic Curves Over a p -adic Field

   https://doi.org/10.1093/imrn/rnab020  

   Gazaki, Evangelia; Leal, Isabel (March 2021, International Mathematics Research Notices)

   Abstract We consider a product $$X=E_1\times \cdots \times E_d$$ of elliptic curves over a finite extension $$K$$ of $${\mathbb{Q}}_p$$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $$X$$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $$CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$$. We give specific criteria that guarantee this map is injective for every $$n\geq 1$$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $$L$$ of $$K$$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi. 

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5. On the Fine-Grained Complexity of Empirical Risk Minimization: Kernel Methods and Neural Networks

   Backurs, Arturs; Indyk, Piotr; Schmidt, Ludwig (January 2017, Annual Conference on Neural Information Processing Systems)

   Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods. While there is a large body of work on algorithms for various ERM problems, the exact computational complexity of ERM is still not understood. We address this issue for multiple popular ERM problems including kernel SVMs, kernel ridge regression, and training the final layer of a neural network. In particular, we give conditional hardness results for these problems based on complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. Under these assumptions, we show that there are no algorithms that solve the aforementioned ERM problems to high accuracy in sub-quadratic time. We also give similar hardness results for computing the gradient of the empirical loss, which is the main computational burden in many non-convex learning tasks. 

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