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Noga Alon

Publications and source records attributed to Noga Alon.

At least 55 records · Page 3Linked to original sources
arXiv2023

Connectivity Graph-Codes

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$ of spanning subgraphs of $H$ a connectivity code for $H$ if the symmetric difference of any two distinct subgraphs in ${\cal G}$ is a connected spanning subgraph of $H$. It is easy to see that the maximum possible cardinality of such a collection is at most $2^{k'(H)} \leq 2^{δ(H)}$, where $k'(H)$ is the edge-connectivity of $H$ and $δ(H)$ is its minimum degree. We show that equality holds for any $d$-regular (mild) expander, and observe that equality does not hold in several natural examples including any large cubic graph, the square of a long cycle and products of a small clique with a long cycle.

math.CO↗
arXiv2023

The success probability in Levine's hat problem, and independent sets in graphs

Lionel Levine's hat challenge has $t$ players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture is that the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.

math.CO↗
arXiv2023

Ordering Candidates via Vantage Points

Given an $n$-element set $C\subseteq\mathbb{R}^d$ and a (sufficiently generic) $k$-element multiset $V\subseteq\mathbb{R}^d$, we can order the points in $C$ by ranking each point $c\in C$ according to the sum of the distances from $c$ to the points of $V$. Let $Ψ_k(C)$ denote the set of orderings of $C$ that can be obtained in this manner as $V$ varies, and let $ψ^{\mathrm{max}}_{d,k}(n)$ be the maximum of $\lvertΨ_k(C)\rvert$ as $C$ ranges over all $n$-element subsets of $\mathbb{R}^d$. We prove that $ψ^{\mathrm{max}}_{d,k}(n)=Θ_{d,k}(n^{2dk})$ when $d \geq 2$ and that $ψ^{\mathrm{max}}_{1,k}(n)=Θ_k(n^{4\lceil k/2\rceil -1})$. As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set $Ψ(C)=\bigcup_{k\geq 1}Ψ_k(C)$; this includes an exact description of $Ψ(C)$ when $d=1$ and when $C$ is the set of vertices of a vertex-transitive polytope.

math.CO↗
arXiv2023

Sublinear Time Shortest Path in Expander Graphs

Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.

cs.DS↗
arXiv2023

On bipartite coverings of graphs and multigraphs

A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this note we establish a (modest) strengthening of old results of Hansel and of Katona and Szemerédi, by showing that the capacity of any bipartite covering of a graph on $n$ vertices in which the maximum size of an independent set containing vertex number $i$ is $α_i$, is at least $\sum_i \log_2 (n/α_i).$ We also obtain slightly improved bounds for a recent result of Kim and Lee about the minimum possible capacity of a bipartite covering of complete multigraphs.

math.CO↗
arXiv2023

On sums and products along the edges, II

This note is a continuation of an earlier paper by the authors. We describe improved constructions addressing a question of Erdős and Szemerédi on sums and products of real numbers along the edges of a graph. We also add a few observations about related versions of the problem.

math.CO↗
arXiv2023

Boosting Simple Learners

Boosting is a celebrated machine learning approach which is based on the idea of combining weak and moderately inaccurate hypotheses to a strong and accurate one. We study boosting under the assumption that the weak hypotheses belong to a class of bounded capacity. This assumption is inspired by the common convention that weak hypotheses are "rules-of-thumbs" from an "easy-to-learn class". (Schapire and Freund~'12, Shalev-Shwartz and Ben-David '14.) Formally, we assume the class of weak hypotheses has a bounded VC dimension. We focus on two main questions: (i) Oracle Complexity: How many weak hypotheses are needed to produce an accurate hypothesis? We design a novel boosting algorithm and demonstrate that it circumvents a classical lower bound by Freund and Schapire ('95, '12). Whereas the lower bound shows that $Ω({1}/{γ^2})$ weak hypotheses with $γ$-margin are sometimes necessary, our new method requires only $\tilde{O}({1}/γ)$ weak hypothesis, provided that they belong to a class of bounded VC dimension. Unlike previous boosting algorithms which aggregate the weak hypotheses by majority votes, the new boosting algorithm uses more complex ("deeper") aggregation rules. We complement this result by showing that complex aggregation rules are in fact necessary to circumvent the aforementioned lower bound. (ii) Expressivity: Which tasks can be learned by boosting weak hypotheses from a bounded VC class? Can complex concepts that are "far away" from the class be learned? Towards answering the first question we {introduce combinatorial-geometric parameters which capture expressivity in boosting.} As a corollary we provide an affirmative answer to the second question for well-studied classes, including half-spaces and decision stumps. Along the way, we establish and exploit connections with Discrepancy Theory.

cs.LG↗
arXiv2023

Strong blocking sets and minimal codes from expander graphs

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the $(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k )$. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every prime power $q$, for which $n = O (q k)$. This solves one of the main open problems on minimal codes.

math.CO↗
arXiv2023

New bounds on the maximum number of neighborly boxes in R^d

A family of axis-aligned boxes in $\er^d$ is \emph{$k$-neighborly} if the intersection of every two of them has dimension at least $d-k$ and at most $d-1$. Let $n(k,d)$ denote the maximum size of such a family. It is known that $n(k,d)$ can be equivalently defined as the maximum number of vertices in a complete graph whose edges can be covered by $d$ complete bipartite graphs, with each edge covered at most $k$ times. We derive a new upper bound on $n(k,d)$, which implies, in particular, that $n(k,d)\leqslant (2-δ)^d$ if $k\leqslant (1-\varepsilon)d$, where $δ>0$ depends on arbitrarily chosen $\varepsilon>0$. The proof applies a classical result of Kleitman, concerning the maximum size of sets with a given diameter in discrete hypercubes. By an explicit construction we obtain also a new lower bound for $n(k,d)$, which implies that $n(k,d)\geqslant (1-o(1))\frac{d^k}{k!}$. We also study $k$-neighborly families of boxes with additional structural properties. Families called \emph{total laminations}, that split in a tree-like fashion, turn out to be particularly useful for explicit constructions. We pose a few conjectures based on these constructions and some computational experiments.

math.CO↗
arXiv2023

Graph-codes

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $[n]=\{1,2, \ldots ,n\}$ is the graph on $[n]$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. Let $H$ be a fixed graph with an even (positive) number of edges, and let $D_H(n)$ denote the maximum possible cardinality of a family of graphs on $[n]$ containing no two members whose symmetric difference is a copy of $H$. Is it true that $D_H(n)=o(2^{n \choose 2})$ for any such $H$? We discuss this problem, compute the value of $D_H(n)$ up to a constant factor for stars and matchings, and discuss several variants of the problem including ones that have been considered in earlier work.

math.CO↗
arXiv2023

Hitting a prime in 2.43 dice rolls (on average)

What is the number of rolls of fair 6-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10^{-4}. This is a solution to a puzzle suggested by DasGupta (2017) in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes.

math.PR↗
arXiv2023

Diagonalization Games

We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a "scientific charlatan". In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors but he is allowed to ask queries of the form"What is bit number j of vector number i?" What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if he is allowed to pick his queries \emph{adaptively}, based on Kronecker's previous replies? The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower bounds in the non-adaptive case.

math.CO↗
arXiv2022

EFX Allocations: Simplifications and Improvements

The existence of EFX allocations is a fundamental open problem in discrete fair division. Given a set of agents and indivisible goods, the goal is to determine the existence of an allocation where no agent envies another following the removal of any single good from the other agent's bundle. Since the general problem has been illusive, progress is made on two fronts: $(i)$ proving existence when the number of agents is small, $(ii)$ proving existence of relaxations of EFX. In this paper, we improve results on both fronts (and simplify in one of the cases). We prove the existence of EFX allocations with three agents, restricting only one agent to have an MMS-feasible valuation function (a strict generalization of nice-cancelable valuation functions introduced by Berger et al. which subsumes additive, budget-additive and unit demand valuation functions). The other agents may have any monotone valuation functions. Our proof technique is significantly simpler and shorter than the proof by Chaudhury et al. on existence of EFX allocations when there are three agents with additive valuation functions and therefore more accessible. Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX allocations and EFX allocations with few unallocated goods (charity). Chaudhury et al. showed the existence of $(1-ε)$-EFX allocation with $O((n/ε)^{\frac{4}{5}})$ charity by establishing a connection to a problem in extremal combinatorics. We improve their result and prove the existence of $(1-ε)$-EFX allocations with $\tilde{O}((n/ ε)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be used to prove improved upper-bounds on a problem in zero-sum combinatorics introduced by Alon and Krivelevich.

cs.GT↗
arXiv2022

Counting Dope Matrices

For a polynomial $P$ of degree $n$ and an $m$-tuple $Λ=(λ_1,\dots,λ_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $Λ$ is $D_P(Λ)=(δ_{ij})_{i\in [1,m],j\in[0,n]}$, where $δ_{ij}=1$ if $P^{(j)}(λ_i)=0$, and $δ_{ij}=0$ otherwise. Our first result is a combinatorial characterization of the $2$-row dope matrices (for all pairs $Λ$); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of $m\times(n+1)$ dope matrices, and we show that the number of $m \times (n+1)$ dope matrices for a fixed $m$-tuple $Λ$ is maximized when $Λ$ is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems.

math.CO↗
arXiv2022

Cats in cubes

Answering a recent question of Patchell and Spiro, we show that when a $d$-dimensional cube of side length $n$ is filled with letters, the word $\mathsf{CAT}$ can appear contiguously at most $(3^{d-1}/2)n^d$ times (allowing diagonals); we also characterize when equality occurs and extend our results to words other than $\mathsf{CAT}$.

math.CO↗
arXiv2022

Turán graphs with bounded matching number

We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.

math.CO↗
arXiv2022

Largest subgraph from a hereditary property in a random graph

We prove that for every non-trivial hereditary family of graphs ${\cal P}$ and for every fixed $p \in (0,1)$, the maximum possible number of edges in a subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high probability, $$ \left(1-\frac{1}{k-1}+o(1)\right)p{n \choose 2}, $$ where $k$ is the minimum chromatic number of a graph that does not belong to ${\cal P}$.

math.CO↗