arXiv subjects

Noga Alon

Publications and source records attributed to Noga Alon.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Bicriterion Concentration Inequality and Prophet Inequalities for $k$-Fold Matroid Unions

We investigate prophet inequalities with competitive ratios approaching $1$, seeking to generalize $k$-uniform matroids. We first show that large girth does not suffice: for all $k$, there exists a matroid of girth $\geq k$ and a prophet inequality instance on that matroid whose optimal competitive ratio is $\frac{1}{2}$. Next, we show $k$-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio $1-O(\sqrt{\frac{\log k}{k}})$ for any $k$-fold matroid union. Our prophet inequality follows from an online contention resolution scheme. The key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone $1$-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a $1$-Lipschitz function that is not (approximately) self-bounding.

cs.DS↗

arXiv2024

Unit and distinct distances in typical norms

Erdős' unit distance problem and Erdős' distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of distinct distances, respectively, determined by $n$ points in the Euclidean plane. The question of what happens in these problems if one considers normed spaces other than the Euclidean plane has been raised in the 1980s by Ulam and Erdős and attracted a lot of attention over the years. We give an essentially tight answer to both questions for almost all norms on $\mathbb{R}^d$, in a certain Baire categoric sense. For the unit distance problem we prove that for almost all norms $\|.\|$ on $\mathbb{R}^d$, any set of $n$ points defines at most $\frac{1}{2} d \cdot n \log_2 n$ unit distances according to $\|.\|$. We also show that this is essentially tight, by proving that for every norm $\|.\|$ on $\mathbb{R}^d$, for any large $n$, we can find $n$ points defining at least $\frac{1}{2}(d-1-o(1))\cdot n \log_2 n$ unit distances according to $\|.\|$. For the distinct distances problem, we prove that for almost all norms $\|.\|$ on $\mathbb{R}^d$ any set of $n$ points defines at least $(1-o(1))n$ distinct distances according to $\|.\|$. This is clearly tight up to the $o(1)$ term. We also answer the famous Hadwiger--Nelson problem for almost all norms on $\mathbb{R}^2$, showing that their unit distance graph has chromatic number $4$. Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matoušek, Brass-Moser-Pach, Chilakamarri, and Robertson. The proofs combine combinatorial and geometric ideas with tools from Linear Algebra, Topology and Algebraic Geometry.

math.CO↗

arXiv2024

Partitioning the hypercube into smaller hypercubes

Denote by Q_d the d-dimensional hypercube. Addressing a recent question we estimate the number of ways the vertex set of Q_d can be partitioned into vertex disjoint smaller cubes. Among other results, we prove that the asymptotic order of this function is not much larger than the number of perfect matchings of Q_d. We also describe several new (and old) questions.

math.CO↗

arXiv2024

Maximum shattering

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum possible number of subsets of $[n]$ of size $d$ that can be shattered by a family of size $k$. Among other results, we determine $f(n,k,d)$ exactly for $d \leq 2$ and show that if $d$ and $n$ grow, with both $d$ and $n-d$ tending to infinity, then, for any $k$ satisfying $2^d \leq k \leq (1+o(1))2^d$, we have $f(n,k,d)=(1+o(1))c\binom{n}{d}$, where $c$, roughly $0.289$, is the probability that a large square matrix over $\mathbb{F}_2$ is invertible. This latter result extends work of Das and Mészáros. As an application, we improve bounds for the existence of covering arrays for certain alphabet sizes.

math.CO↗

arXiv2024

On hypercube statistics

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $λ(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let $λ(n,d,s)$ denote the maximum possible value of $λ(n,d,s,A)$ as $A$ ranges over all subsets of vertices of $Q_n$, and let $λ(d,s)$ denote the limit of this quantity as $n$ tends to infinity. We prove several lower and upper bounds on $λ(d,s)$, showing that for all admissible values of $d$ and $s$ it is larger than $0.28$. We also show that the values of $s=s(d)$ such that $λ(d,s)=1$ are exactly $\{0,2^{d-1},2^d\}$. In addition we prove that if $0<s< d/8$, then $λ(d, s) \leq 1 - Ω(1/s)$, and that if $s$ is divisible by a power of $2$ which is $Ω(s)$ then $λ(d,s) \geq 1-O(1/s)$. We suspect that $λ(d,1)=(1+o(1))/e$ where the $o(1)$-term tends to $0$ as $d$ tends to infinity, but this remains open, as does the problem of obtaining tight bounds for essentially all other quantities $λ(d,s)$.

math.CO↗

arXiv2024

A Unified Characterization of Private Learnability via Graph Theory

We provide a unified framework for characterizing pure and approximate differentially private (DP) learnability. The framework uses the language of graph theory: for a concept class $\mathcal{H}$, we define the contradiction graph $G$ of $\mathcal{H}$. Its vertices are realizable datasets, and two datasets $S,S'$ are connected by an edge if they contradict each other (i.e., there is a point $x$ that is labeled differently in $S$ and $S'$). Our main finding is that the combinatorial structure of $G$ is deeply related to learning $\mathcal{H}$ under DP. Learning $\mathcal{H}$ under pure DP is captured by the fractional clique number of $G$. Learning $\mathcal{H}$ under approximate DP is captured by the clique number of $G$. Consequently, we identify graph-theoretic dimensions that characterize DP learnability: the clique dimension and fractional clique dimension. Along the way, we reveal properties of the contradiction graph which may be of independent interest. We also suggest several open questions and directions for future research.

cs.LG↗

arXiv2024

Optimal Preprocessing for Answering On-Line Product Queries

We examine the amount of preprocessing needed for answering certain on-line queries as fast as possible. We start with the following basic problem. Suppose we are given a semigroup $(S,\circ )$. Let $s_1 ,\ldots, s_n$ be elements of $S$. We want to answer on-line queries of the form, ``What is the product $s_i \circ s_{i+1} \circ \cdots \circ s_{j-1} \circ s_j$?'' for any given $1\le i\le j\le n$. We show that a preprocessing of $Θ(n λ(k,n))$ time and space is both necessary and sufficient to answer each such query in at most $k$ steps, for any fixed $k$. The function $λ(k,\cdot)$ is the inverse of a certain function at the $\lfloor {k/2}\rfloor$-th level of the primitive recursive hierarchy. In case linear preprocessing is desired, we show that one can answer each such query in $O( α(n))$ steps and that this is best possible. The function $α(n)$ is the inverse Ackermann function. We also consider the following extended problem. Let $T$ be a tree with an element of $S$ associated with each of its vertices. We want to answer on-line queries of the form, ``What is the product of the elements associated with the vertices along the path from $u$ to $v$?'' for any pair of vertices $u$ and $v$ in $T$. We derive results that are similar to the above, for the preprocessing needed for answering such queries. All our sequential preprocessing algorithms can be parallelized efficiently to give optimal parallel algorithms which run in $O(\log n)$ time on a CREW PRAM. These parallel algorithms are optimal in both running time and total number of operations. Our algorithms, especially for the semigroup of the real numbers with the minimum or maximum operations, have various applications in certain graph algorithms, in the utilization of communication networks and in Database retrieval.

cs.DS↗

arXiv2024

The power of many colours

A classical problem, due to Gerencsér and Gyárfás from 1967, asks how large a monochromatic connected component can we guarantee in any $r$-edge colouring of $K_n$? We consider how big a connected component can we guarantee in any $r$-edge colouring of $K_n$ if we allow ourselves to use up to $s$ colours. This is actually an instance of a more general question of Bollobás from about 20 years ago which asks for a $k$-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided $n$ is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by Erdős and Rényi in 1956, who considered given $n$ and $m$, what is the smallest number of $m$-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any $r$-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to $s$ colours. We obtain a more complete understanding of the answer to this question for large $n$, in particular determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

math.CO↗

arXiv2024

The Helly number of Hamming balls and related problems

We prove the following variant of Helly's classical theorem for Hamming balls with a bounded radius. For $n>t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X^n$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X|>1$ and all $n>t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory.

math.CO↗

arXiv2024

Rainbow Stackings of Random Edge-Colorings

A rainbow stacking of $r$-edge-colorings $χ_1, \ldots, χ_m$ of the complete graph on $n$ vertices is a way of superimposing $χ_1, \ldots, χ_m$ so that no edges of the same color are superimposed on each other. We determine a sharp threshold for $r$ (as a function of $m$ and $n$) governing the existence and nonexistence of rainbow stackings of random $r$-edge-colorings $χ_1,\ldots,χ_m$.

math.CO↗

arXiv2024

Sumsets in the Hypercube

A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically $(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in $\mathbb{F}_2^n$ is almost identical to the family of all subsets of $\mathbb{F}_2^n$ that contain a complete linear subspace of co-dimension $1$.

math.CO↗

arXiv2024

Identifying the Deviator

A group of players are supposed to follow a prescribed profile of strategies. If they follow this profile, they will reach a given target. We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.

math.PR↗

arXiv2024

Erasure codes and Turán hypercube problems

We observe that several vertex Turán type problems for the hypercube that received a considerable amount of attention in the combinatorial community are equivalent to questions about erasure list-decodable codes. Analyzing a recent construction of Ellis, Ivan and Leader, and determining the Turán density of certain hypergraph augemntations we obtain improved bounds for some of these problems.

math.CO↗

arXiv2024

Invertibility of digraphs and tournaments

For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the digraph obtained by reversing the orientations of the edges of $D$ with both endpoints in $X$. The inversion number of $D$, $\textrm{inv}(D)$, is the minimum number of inversions which can be applied in turn to $D$ to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each $k\in\mathbb{N}$ and tournament $T$, the problem of deciding whether $\textrm{inv}(T)\leq k$ is solvable in time $O_k(|V(T)|^2)$, which is tight for all $k$. In particular, the problem is fixed-parameter tractable when parameterised by $k$. On the other hand, we build on their work to prove their conjecture that for $k\geq 1$ the problem of deciding whether a general oriented graph $D$ has $\textrm{inv}(D)\leq k$ is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called 'dijoin' digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an $n$-vertex tournament is $(1+o(1))n$.

math.CO↗

arXiv2024

Universality for graphs with bounded density

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an $\mathcal{H}$-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil + 1)} \right)$ edges which contains every $n$-vertex graph with density at most $d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $Ω(n^{2 - 1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain a near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an asymptotically optimal construction.

math.CO↗

arXiv2023

On a random model of forgetting

Georgiou, Katkov and Tsodyks considered the following random process. Let $x_1,x_2,\ldots $ be an infinite sequence of independent, identically distributed, uniform random points in $[0,1]$. Starting with $S=\{0\}$, the elements $x_k$ join $S$ one by one, in order. When an entering element is larger than the current minimum element of $S$, this minimum leaves $S$. Let $S(1,n)$ denote the content of $S$ after the first $n$ elements $x_k$ join. Simulations suggest that the size $|S(1,n)|$ of $S$ at time $n$ is typically close to $n/e$. Here we first give a rigorous proof that this is indeed the case, and that in fact the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ is of size at most $\tilde{O}(\sqrt n)$ with high probability. Our main result is a more accurate description of the process implying, in particular, that as $n$ tends to infinity $ n^{-1/2}\big( |S(1,n)|-n/e \big) $ converges to a normal random variable with variance $3e^{-2}-e^{-1}$. We further show that the dynamics of the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ converges with proper scaling to a three dimensional Bessel process.

math.PR↗

arXiv2023

Optimal Sample Complexity of Contrastive Learning

Contrastive learning is a highly successful technique for learning representations of data from labeled tuples, specifying the distance relations within the tuple. We study the sample complexity of contrastive learning, i.e. the minimum number of labeled tuples sufficient for getting high generalization accuracy. We give tight bounds on the sample complexity in a variety of settings, focusing on arbitrary distance functions, both general $\ell_p$-distances, and tree metrics. Our main result is an (almost) optimal bound on the sample complexity of learning $\ell_p$-distances for integer $p$. For any $p \ge 1$ we show that $\tilde Θ(\min(nd,n^2))$ labeled tuples are necessary and sufficient for learning $d$-dimensional representations of $n$-point datasets. Our results hold for an arbitrary distribution of the input samples and are based on giving the corresponding bounds on the Vapnik-Chervonenkis/Natarajan dimension of the associated problems. We further show that the theoretical bounds on sample complexity obtained via VC/Natarajan dimension can have strong predictive power for experimental results, in contrast with the folklore belief about a substantial gap between the statistical learning theory and the practice of deep learning.

cs.LG↗

arXiv2023

The limit points of the top and bottom eigenvalues of regular graphs

We prove that for each $d \geq 3$ the set of all limit points of the second largest eigenvalue of growing sequences of $d$-regular graphs is $[2\sqrt{d-1},d]$. A similar argument shows that the set of all limit points of the smallest eigenvalue of growing sequences of $d$-regular graphs with growing (odd) girth is $[-d, -2 \sqrt{d-1}]$. The more general question of identifying all vectors which are limit points of the vectors of the top $k$ eigenvalues of sequences of $d$-regular graphs is considered as well. As a by product, in the study of discrete counterpart of the "scarring" phenomenon observed in the investigation of quantum ergodicity on manifolds, our technique provides a method to construct $d$-regular almost Ramanujan graphs with large girth and localized eigenvectors corresponding to eigenvalues larger than $2\sqrt{d-1}$, strengthening a result of Alon, Ganguly, and Srivastava.

math.CO↗