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1. Online Spanners in Metric Spaces

   Bhore, Sujoy ; Filtser, Arnold ; Khodabandeh, Hadi ; Toth, Csaba D. ( September 2022 , 30th Annual European Symposium on Algorithms (ESA 2022))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

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2. Online Spanners in Metric Spaces

   Bhore, Sujoy ; Filtser, Arnold ; Khodabandeh, Hadi ; Toth, Csaba D. ( September 2022 , Proc. 30th Annual European Symposium on Algorithms (ESA))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

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3. Approximate shortest paths and distance oracles in weighted unit-disk graphs

   https://doi.org/10.20382/jocg.v10i2  

   Chan, Timothy M. ; Skrepetos, Dimitrios ( January 2019 , Journal of computational geometry)

   We present the first near-linear-time algorithm that computes a (1+ε)-approximation of the diameter of a weighted unit-disk graph of n vertices. Our algorithm requires O(n log^2 n) time for any constant ε>0, so we considerably improve upon the near-O(n^{3/2})-time algorithm of Gao and Zhang (2005). Using similar ideas we develop (1+ε)-approximate \emph{distance oracles} of O(1) query time with a likewise improvement in the preprocessing time, specifically from near O(n^{3/2}) to O(n log^3 n). We also obtain similar new results for a number of related problems in the weighted unit-disk graph metric such as the radius and the bichromatic closest pair. As a further application we employ our distance oracle, along with additional ideas, to solve the (1+ε)-approximate \emph{all-pairs bounded-leg shortest paths\/} (apBLSP) problem for a set of n planar points. Our data structure requires O(n^2 log n) space, O(loglog n) query time, and nearly O(n^{2.579}) preprocessing time for any constant ε>0, and is the first that breaks the near-cubic preprocessing time bound given by Roditty and Segal (2011). 

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4. Simple and efficient pseudorandom genera- tors from Gaussian processes.

   Chattopadhyay, Eshan ; De, Anindya ; Servedio, Rocco ( July 2019 , Leibniz international proceedings in informatics)

   Abstract We show that a very simple pseudorandom generator fools intersections of k linear threshold functions (LTFs) and arbitrary functions of k LTFs over n-dimensional Gaussian space. The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of PRGs for functions of halfspaces. Our analysis for arbitrary functions of LTFs establishes bounds on the Wasserstein distance between Gaussian random vectors with similar covariance matrices, and combines these bounds with a conversion from Wasserstein distance to "union-of-orthants" distance from [Xi Chen et al., 2014]. Our analysis for intersections of LTFs uses extensions of the classical Sudakov-Fernique type inequalities, which give bounds on the difference between the expectations of the maxima of two Gaussian random vectors with similar covariance matrices. For all values of k, our generator has seed length O(log n) + poly(k) for arbitrary functions of k LTFs and O(log n) + poly(log k) for intersections of k LTFs. The best previous result, due to [Gopalan et al., 2010], only gave such PRGs for arbitrary functions of k LTFs when k=O(log log n) and for intersections of k LTFs when k=O((log n)/(log log n)). Thus our PRG achieves an O(log n) seed length for values of k that are exponentially larger than previous work could achieve. By combining our PRG over Gaussian space with an invariance principle for arbitrary functions of LTFs and with a regularity lemma, we obtain a deterministic algorithm that approximately counts satisfying assignments of arbitrary functions of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(k,1/epsilon)} for all values of k. This algorithm has a poly(n) runtime for k =(log n)^c for some absolute constant c>0, while the previous best poly(n)-time algorithms could only handle k = O(log log n). For intersections of LTFs, by combining these tools with a recent PRG due to [R. O'Donnell et al., 2018], we obtain a deterministic algorithm that can approximately count satisfying assignments of intersections of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(log k, 1/epsilon)}. This algorithm has a poly(n) runtime for k =2^{(log n)^c} for some absolute constant c>0, while the previous best poly(n)-time algorithms for intersections of k LTFs, due to [Gopalan et al., 2010], could only handle k=O((log n)/(log log n)). 

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5. From approximate to exact integer programming

   https://doi.org/10.1007/s10107-024-02084-1  

   Dadush, Daniel ; Eisenbrand, Friedrich ; Rothvoss, Thomas ( April 2024 , Mathematical Programming)

   Approximate integer programming is the following: For a given convex body$$K \subseteq {\mathbb {R}}^n$$$K\subseteq R^n$, either determine whether$$K \cap {\mathbb {Z}}^n$$$K\cap Z^n$is empty, or find an integer point in the convex body$$2\cdot (K - c) +c$$$2·(K-c)+c$which isK, scaled by 2 from its center of gravityc. Approximate integer programming can be solved in time$$2^{O(n)}$$$2^{O\left(n\right)}$while the fastest known methods for exact integer programming run in time$$2^{O(n)} \cdot n^n$$$2^{O\left(n\right)}·n^n$. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$x^* \in (K \cap {\mathbb {Z}}^n)$$$x^{\ast }\in (K\cap Z^n)$can be found in time$$2^{O(n)}$$$2^{O\left(n\right)}$, provided that theremaindersof each component$$x_i^* \mod \ell $$$x_i^{\ast }mod\ell$for some arbitrarily fixed$$\ell \ge 5(n+1)$$$\ell \ge 5(n+1)$of$$x^*$$$x^{\ast }$are given. The algorithm is based on acutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a$$2^{O(n)}n^n$$$2^{O\left(n\right)}{n}^n$algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a newasymmetric approximate Carathéodory theoremthat might be of interest on its own. Our second method concerns integer programming problems in equation-standard form$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$$Ax=b,0\le x\le u,x\in Z^n$. Such a problem can be reduced to the solution of$$\prod _i O(\log u_i +1)$$${\prod }_{i}O(log{u}_i+1)$approximate integer programming problems. This implies, for example thatknapsackorsubset-sumproblems withpolynomial variable range$$0 \le x_i \le p(n)$$$0\le x_i\le p\left(n\right)$can be solved in time$$(\log n)^{O(n)}$$${(logn)}^{O\left(n\right)}$. For these problems, the best running time so far was$$n^n \cdot 2^{O(n)}$$$n^n·2^{O\left(n\right)}$.

    

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