arXiv subjects

Nati Linial

Publications and source records attributed to Nati Linial.

At least 37 records · Page 2Linked to original sources

arXiv2022

On the local structure of oriented graphs -- a case study in flag algebras

Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be the probability that a random set of $3$ vertices of $G$ spans a transitive triangle (respectively an independent set). We prove that $t(G) + i(G) \geq \frac{1}{9}-o_n(1)$. Our proof uses the method of flag algebras that we supplement with several steps that make it more easily comprehensible. We also prove a stability result and an exact result. Namely, we describe an extremal construction, prove that it is essentially unique, and prove that if $H$ is sufficiently far from that construction, then $t(H) + i(H)$ is significantly larger than $\frac{1}{9}$. We go to greater technical detail than is usually done in papers that rely on flag algebras. Our hope is that as a result this text can serve others as a useful introduction to this powerful and beautiful method.

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arXiv2022

An approach to the girth problem in cubic graphs

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph theorem, $E$ is the disjoint union of a $2$-factor and a perfect matching $M$. We refer to the edges of $M$ as chords and classify the cycles in $G$ by their number of chords. We define $γ_k(n)$ to be the largest integer $g$ such that every cubic $n$-vertex graph with a given perfect matching $M$ has a cycle of length at most $g$ with at most $k$ chords. Here we determine this function up to small additive constant for $k= 1, 2$ and up to a small multiplicative constant for larger $k$.

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arXiv2021

Larger Corner-Free Sets from Better NOF Exactly-$N$ Protocols

A subset of the integer planar grid $[N] \times [N]$ is called corner-free if it contains no triple of the form $(x,y), (x+δ,y), (x,y+δ)$. It is known that such a set has a vanishingly small density, but how large this density can be remains unknown. The best previous construction was based on Behrend's large subset of $[N]$ with no $3$-term arithmetic progression. Here we provide the first substantial improvement to this lower bound in decades. Our approach to the problem is based on the theory of communication complexity. In the $3$-players exactly-$N$ problem the players need to decide whether $x+y+z=N$ for inputs $x,y,z$ and fixed $N$. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Despite the basic nature of this problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-$N$ problem. This is also the first significant example where algorithmic ideas in communication complexity bear fruit in additive combinatorics.

math.CO↗

arXiv2021

Irreducible Non-Metrizable Path Systems in Graphs

A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$ restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we construct an infinite family of non-metrizable irreducible path systems defined on certain Paley graphs.

math.CO↗

arXiv2020

Hyperpaths

Hypertrees are high-dimensional counterparts of graph theoretic trees. They have attracted a great deal of attention by various investigators. Here we introduce and study Hyperpaths -- a particular class of hypertrees which are high dimensional analogs of paths in graph theory. A $d$-dimensional hyperpath is a $d$-dimensional hypertree in which every $(d-1)$-dimensional face is contained in at most $(d+1)$ faces of dimension $d$. We introduce a possibly infinite family of hyperpaths for every dimension, and investigate its properties in greater depth for dimension $d=2$.

math.CO↗

arXiv2020

Geodesic Geometry on Graphs

We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $\mathcal{P}=\{P_{u,v}|u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y, z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w: E\to(0,\infty)$ is said to induce $\mathcal{P}$ if for every $u, v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many $2$-connected metrizable graphs.

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arXiv2020

A randomized construction of high girth regular graphs

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices. As long as the smallest degree $δ(G)<k$, choose, uniformly at random, two vertices $u,v \in V(G)$ of degree $δ(G)$ whose distance is at least $g-1$. If there are no such vertex pairs, abort. Otherwise, add the edge $uv$ to $E(G)$. We show that with high probability this algorithm yields a $k$-regular graph with girth at least $g$. Our analysis also implies that there are $\left( Ω(n) \right)^{kn/2}$ labeled $k$-regular $n$-vertex graphs with girth at least $g$.

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arXiv2019

A King in every two consecutive tournaments

We think of a tournament $T=([n], E)$ as a communication network where in each round of communication processor $P_i$ sends its information to $P_j$, for every directed edge $ij \in E(T)$. By Landau's theorem (1953) there is a King in $T$, i.e., a processor whose initial input reaches every other processor in two rounds or less. Namely, a processor $P_ν$ such that after two rounds of communication along $T$'s edges, the initial information of $P_ν$ reaches all other processors. Here we consider a more general scenario where an adversary selects an arbitrary series of tournaments $T_1, T_2,\ldots$, so that in each round $s=1, 2, \ldots$, communication is governed by the corresponding tournament $T_s$. We prove that for every series of tournaments that the adversary selects, it is still true that after two rounds of communication, the initial input of at least one processor reaches everyone. Concretely, we show that for every two tournaments $T_1, T_2$ there is a vertex in $[n]$ that can reach all vertices via (i) A step in $T_1$, or (ii) A step in $T_2$ or (iii) A step in $T_1$ followed by a step in $T_2$. }

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arXiv2019

Expander Graphs -- Both Local and Global

Let $G=(V,E)$ be a finite graph. For $v\in V$ we denote by $G_v$ the subgraph of $G$ that is induced by $v$'s neighbor set. We say that $G$ is $(a,b)$-regular for $a>b>0$ integers, if $G$ is $a$-regular and $G_v$ is $b$-regular for every $v\in V$. Recent advances in PCP theory call for the construction of infinitely many $(a,b)$-regular expander graphs $G$ that are expanders also locally. Namely, all the graphs $\{G_v|v\in V\}$ should be expanders as well. While random regular graphs are expanders with high probability, they almost surely fail to expand locally. Here we construct two families of $(a,b)$-regular graphs that expand both locally and globally. We also analyze the possible local and global spectral gaps of $(a,b)$-regular graphs. In addition, we examine our constructions vis-a-vis properties which are considered characteristic of high-dimensional expanders.

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arXiv2018

On The Communication Complexity of High-Dimensional Permutations

We study the multiparty communication complexity of high dimensional permutations, in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer $n$. There is a considerable body of literature dealing with the same problem, where $(\mathbb{N},+)$ is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of work. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that appeal to recent advances in Additive Combinatorics and Ramsey theory. We reveal new and unexpected connections between the NOF communication complexity of high dimensional permutations and a variety of well known and thoroughly studied problems in combinatorics. Previous protocols for Exactly-n all rely on the construction of large sets of integers without a 3-term arithmetic progression. No direct algorithmic protocol was previously known for the problem, and we provide the first such algorithm. This suggests new ways to significantly improve the CFL protocol. Many new open questions are presented throughout.

cs.CC↗

arXiv2018

Universal Knot Diagrams

We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.

math.GT↗

arXiv2018

On the weight distribution of random binary linear codes

We investigate the weight distribution of random binary linear codes. For $0<λ<1$ and $n\to\infty$ pick uniformly at random $λn$ vectors in $\mathbb{F}_2^n$ and let $C \le \mathbb{F}_2^n$ be the orthogonal complement of their span. Given $0<γ<1/2$ with $0< λ< h(γ)$ let $X$ be the random variable that counts the number of words in $C$ of Hamming weight $γn$. In this paper we determine the asymptotics of the moments of $X$ of all orders $o(\frac{n}{\log n})$.

cs.IT↗

arXiv2018

The Distribution of Knots in the Petaluma Model

The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.

math.GT↗

arXiv2018

Enumeration and randomized constructions of hypertrees

Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog of Cayley's formula for the number of $n$-vertex trees. He enumerated $d$-dimensional hypertrees weighted by the squared size of their $(d-1)$-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of $d$-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of $d$-hypertrees. In addition, we study a random $1$-out model of $d$-complexes where every $(d-1)$-dimensional face selects a random $d$-face containing it, and show it has a negligible $d$-dimensional homology.

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arXiv2017

Efficient Generation of One-Factorizations through Hill Climbing

It is well known that for every even integer $n$, the complete graph $K_{n}$ has a one-factorization, namely a proper edge coloring with $n-1$ colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of one-factorizations. Specifically, we know essentially nothing about the {\em typical} properties of one-factorizations for large $n$. Suppose that $\cal C_{\rm n}$ is a graph whose vertex set includes the set of all order-$n$ one-factorizations and that $Ψ: V(\cal C_{\rm n})\to \mathbb R$ takes its minimum precisely at the one-factorizations. Given $\cal C_{\rm n}$ and $Ψ$, we can generate one-factorizations via hill climbing. Namely, by taking a walk on $\cal C_{\rm n}$ that tends to go from a vertex to a neighbor of smaller $Ψ$. For over 30 years, hill-climbing has been essentially the only method for generating many large one-factorizations. However, the validity of such methods was supported so far only by numerical evidence. Here, we present for the first time hill-climbing algorithms that provably generate an order-$n$ one-factorization in $\text{polynomial}(n)$ steps regardless of the starting state, while all vertex degrees in the underlying graph are appropriately bounded. We also raise many questions and conjectures regarding hill-climbing methods and concerning the possible and typical structure of one-factorizations.

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arXiv2016

Random simplicial complexes - around the phase transition

This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in $G(n, p)$ theory that occurs at $p = \frac 1n$ . Our main objective is to provide a more streamlined and unified perspective of some of the papers in this area.

math.CO↗

arXiv2016

Invariants of Random Knots and Links

We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams.

math.GT↗

arXiv2015

Extremal problems on shadows and hypercuts in simplicial complexes

Let $F$ be an $n$-vertex forest. We say that an edge $e\notin F$ is in the shadow of $F$ if $F\cup\{e\}$ contains a cycle. It is easy to see that if $F$ is "almost a tree", that is, it has $n-2$ edges, then at least $\lfloor\frac{n^2}{4}\rfloor$ edges are in its shadow and this is tight. Equivalently, the largest number of edges an $n$-vertex cut can have is $\lfloor\frac{n^2}{4}\rfloor$. These notions have natural analogs in higher $d$-dimensional simplicial complexes, graphs being the case $d=1$. The results in dimension $d>1$ turn out to be remarkably different from the case in graphs. In particular the corresponding bounds depend on the underlying field of coefficients. We find the (tight) analogous theorems for $d=2$. We construct $2$-dimensional "$\mathbb Q$-almost-hypertrees" (defined below) with an empty shadow. We also show that the shadow of an "$\mathbb F_2$-almost-hypertree" cannot be empty, and its least possible density is $Θ(\frac{1}{n})$. In addition we construct very large hyperforests with a shadow that is empty over every field. For $d\ge 4$ even, we construct $d$-dimensional $\mathbb{F} _2$-almost-hypertree whose shadow has density $o_n(1)$. Finally, we mention several intriguing open questions.

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