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cs.DS updates on arXiv.org

PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Large Minors in Expanders
Julia Chuzhoy, Rachit Nimavat · 2019-01-27 · via cs.DS updates on arXiv.org

In this paper we study expander graphs and their minors. Specifically, we attempt to answer the following question: what is the largest function $f(n,α,d)$, such that every $n$-vertex $α$-expander with maximum vertex degree at most $d$ contains {\bf every} graph $H$ with at most $f(n,α,d)$ edges and vertices as a minor? Our main result is that there is some universal constant $c$, such that $f(n,α,d)\geq \frac{n}{c\log n}\cdot \left(\fracα{d}\right )^c$. This bound achieves a tight dependence on $n$: it is well known that there are bounded-degree $n$-vertex expanders, that do not contain any grid with $Ω(n/\log n)$ vertices and edges as a minor. The best previous result showed that $f(n,α,d) \geq Ω(n/\log^κn)$, where $κ$ depends on both $α$ and $d$. Additionally, we provide a randomized algorithm, that, given an $n$-vertex $α$-expander with maximum vertex degree at most $d$, and another graph $H$ containing at most $\frac{n}{c\log n}\cdot \left(\fracα{d}\right )^c$ vertices and edges, with high probability finds a model of $H$ in $G$, in time poly$(n)\cdot (d/α)^{O\left( \log(d/α) \right)}$. We note that similar but stronger results were independently obtained by Krivelevich and Nenadov: they show that $f(n,α,d)=Ω\left(\frac{nα^2}{d^2\log n} \right)$, and provide an efficient algorithm, that, given an $n$-vertex $α$-expander of maximum vertex degree at most $d$, and a graph $H$ with $O\left( \frac{nα^2}{d^2\log n} \right)$ vertices and edges, finds a model of $H$ in $G$. Finally, we observe that expanders are the `most minor-rich' family of graphs in the following sense: for every $n$-vertex and $m$-edge graph $G$, there exists a graph $H$ with $O \left( \frac{n+m}{\log n} \right)$ vertices and edges, such that $H$ is not a minor of $G$.