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1. Fast Approximate Counting of Cycles

   https://doi.org/10.4230/LIPIcs.ICALP.2024.37  

   Censor-Hillel, Keren; Even, Tomer; Vassilevska_Williams, Virginia (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

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

   We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal. 

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2. Large cliques and independent sets all over the place

   https://doi.org/10.1090/proc/15323  

   Alon, Noga; Bucić, Matija; Sudakov, Benny (August 2021, Proceedings of the American Mathematical Society)

   We study the following question raised by Erdős and Hajnal in the early 90’s. Over all n n -vertex graphs G G what is the smallest possible value of m m for which any m m vertices of G G contain both a clique and an independent set of size log ⁡ n \log n ? We construct examples showing that m m is at most 2 2 ( log ⁡ log ⁡ n ) 1 / 2 + o ( 1 ) 2^{2^{(\log \log n)^{1/2+o(1)}}} obtaining a twofold sub-polynomial improvement over the upper bound of about n \sqrt {n} coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size log ⁡ n \log n in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness. 

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3. 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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4. 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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5. 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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