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1. Minimum Weight Euclidean (1+ε)-Spanners

   Toth, Csaba D. (October 2022, Graph-Theoretic Concepts in Computer Science)

   Given a set S of n points in the plane and a parameter ε>0, a Euclidean (1+ε) -spanner is a geometric graph G=(S,E) that contains a path of weight at most (1+ε)∥pq∥2 for all p,q∈S . We show that the minimum weight of a Euclidean (1+ε)-spanner for n points in the unit square [0,1]2 is O(ε−3/2n−−√), and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline O(ε−2n−−√), obtained by combining a tight bound for the weight of an MST and a tight bound for the lightness of Euclidean (1+ε)-spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to d-space for all d∈N : The minimum weight of a Euclidean (1+ε)-spanner for n points in the unit cube [0,1]d is Od(ε(1−d2)/dn(d−1)/d), and this bound is the best possible. For the n×n section of the integer lattice, we show that the minimum weight of a Euclidean (1+ε)-spanner is between Ω(ε−3/4n2) and O(ε−1log(ε−1)n2). These bounds become Ω(ε−3/4n−−√) and O(ε−1log(ε−1)n−−√) when scaled to a grid of n points in [0,1]2. . 

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2. On Euclidean Steiner (1+ε)-Spanners

   https://doi.org/10.4230/LIPIcs.STACS.2021.13  

   Bhore, Sujoy; Toth, Csaba D. (March 2021, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021))

   null (Ed.)

   Lightness and sparsity are two natural parameters for Euclidean (1+ε)-spanners. Classical results show that, when the dimension d ∈ ℕ and ε > 0 are constant, every set S of n points in d-space admits an (1+ε)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ε > 0 for constant d ∈ ℕ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+ε)-spanner. They gave upper bounds of Õ(ε^{-(d+1)/2}) for the minimum lightness in dimensions d ≥ 3, and Õ(ε^{-(d-1))/2}) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+ε)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ε^{-d/2}) for the lightness and Ω(ε^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+ε)-spanners of lightness O(ε^{-1}log n) for n points in Euclidean plane. 

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3. Light Euclidean Steiner Spanners in the Plane

   https://doi.org/10.4230/LIPIcs.SoCG.2021.15  

   Bhore, Sujoy; Toth, Csaba D. (June 2021, 37th International Symposium on Computational Geometry (SoCG 2021))

   null (Ed.)

   Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝ^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ ℕ of the minimum lightness of a (1+ε)-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε^{-(d+1)/2}) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(ε^{-d/2}) for the lightness of Steiner (1+ε)-spanners in ℝ^d, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2. In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1+ε)-spanner of lightness O(ε^{-1}); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis. 

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4. The Bright Side of Simple Heuristics for the TSP

   https://doi.org/10.37236/12651  

   Frieze, Alan; Pegden, Wesley (October 2024, The Electronic Journal of Combinatorics)

   The greedy and nearest-neighbor TSP heuristics can both have $$\log n$$ approximation factors from optimal in worst case, even just for $$n$$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $$d$$-Ahlfor's regular metric space (which includes any $$d$$-manifold like the $$d$$-cube $[0,1]^d$ in the case $$d$$ is an integer but also fractals of dimension $$d$$ when $$d$$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have additive errors from optimal on the order of the optimal tour length through random points in the same space, for $d>1$. 

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5. Approximate Degree, Secret Sharing, and Concentration Phenomena

   https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.71  

   Bogdanov, Andrej; Mande, Nikhil S.; Thaler, Justin; Williamson, Christopher (September 2019, Leibniz international proceedings in informatics)

   The epsilon-approximate degree, deg_epsilon(f), of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are: - We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution. - We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1-norm at least K^{-3/2} * exp ({Omega (deg_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND. 

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