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Publications (73)
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 e...
Let G = (V, E) be a finite graph. For v ∈ V we denote by Gv 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 Gv is b-regular for every v ∈ V. Recent advances in PCP theory call for the construction of infinitely many (a,b)-regular expander graphs G that are expanders al...
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...
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 e...
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 cur...
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...
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 gr...
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. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots. As a corollary, we show that all knots are closures of 1-pure braid...
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...
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 every specific knot decays to zero as n, the number of petals, grows. In a...
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 peta...
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.
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 or...
A triangle-free graph G is called k-existentially complete if for every
induced k-vertex subgraph H of G, every extension of H to a (k+1)-vertex
triangle-free graph can be realized by adding another vertex of G to H. Cherlin
asked whether k-existentially complete triangle-free graphs exist for every k.
Here we present known and new constructions of...
For given integers k, l we ask whether every large graph with a sufficiently
small number of k-cliques and k-anticliques must contain an induced copy of
every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A
similar phenomenon is established as well for tournaments with k=l=4.
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 c...
Market share and quality, or customer satisfaction, go together. Yet
inferring one from the other appears difficult. Indeed, such an inference would
need detailed information about customer behavior, and might be clouded by
modes of behavior such as herding (following popularity) or elitism, where
customers avoid popular products. We investigate a...
For a graph G, let p_i(G), i=0,...,3 be the probability that three distinct
random vertices span exactly i edges. We call (p_0(G),...,p_3(G)) the 3-local
profile of G. We investigate the set ${\cal S}_3 \subset \mathbb R^4$ of all
vectors (p_0,...,p_3) that are arbitrarily close to the 3-local profiles of
arbitrarily large graphs. We give a full de...
If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what
can be said about the number of its $4$-cycles? The most interesting range of
this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for
some $c>0$ and we seek to minimize the number of $4$-cycles. We conjecture that
the (asymptotic) minimizing $T$ is a ran...
There is much recent interest in understanding the density at which constant
size graphs can appear in a very large graph. Specifically, the inducibility of
a graph H is its extremal density, as an induced subgraph of G, where |G| ->
infinity. Already for 4-vertex graphs many questions are still open. Thus, the
inducibility of the 4-path was addres...
We study the local profiles of trees. We show that, in contrast with the
situation for general graphs, the limit set of k-profiles of trees is convex.
We initiate a study of the defining inequalities of this convex set. Many
challenging problems remain open.
An internal partition of an $n$-vertex graph $G=(V,E)$ is a partition of $V$
such that every vertex has at least as many neighbors in its own part as in the
other part. It has been conjectured that every $d$-regular graph with $n>N(d)$
vertices has an internal partition. Here we prove this for $d=6$. The case
$d=n-4$ is of particular interest and l...
We study a high-dimensional analog for the notion of an acyclic (aka
transitive) tournament. We give upper and lower bounds on the number of
$d$-dimensional $n$-vertex acyclic tournaments. In addition, we prove that
every $n$-vertex $d$-dimensional tournament contains an acyclic subtournament
of $\Omega(\log^{1/d}n)$ vertices and the bound is tight...
Let r, s >= 2 be integers. Suppose that the number of blue r-cliques in a
red/blue coloring of the edges of the complete graph K_n is known and fixed.
What is the largest possible number of red s-cliques under this assumption? The
well known Kruskal-Katona theorem answers this question for r=2 or s=2. Using
the shifting technique from extremal set...
Many popular learning algorithms (E.g. Regression, Fourier-Transform based
algorithms, Kernel SVM and Kernel ridge regression) operate by reducing the
problem to a convex optimization problem over a vector space of functions.
These methods offer the currently best approach to several central problems
such as learning half spaces and learning DNF's....
In the {\em Musical Chairs} game $MC(n,m)$ a team of $n$ players plays
against an adversarial {\em scheduler}. The scheduler wins if the game proceeds
indefinitely, while termination after a finite number of rounds is declared a
win of the team. At each round of the game each player {\em occupies} one of
the $m$ available {\em chairs}. Termination...
We consider the following solitaire game whose rules are reminiscent of the childrenʼs game of leapfrog. The game is played on a poset (P,≺)(P,≺) with n elements. The player is handed an arbitrary permutation π=(x1,x2,…,xn)π=(x1,x2,…,xn) of the elements in P. At each round an element may “skip over” a smaller element preceding it, i.e. swap positio...
Numerous papers ask how difficult it is to cluster data. We suggest that the
more relevant and interesting question is how difficult it is to cluster data
sets {\em that can be clustered well}. More generally, despite the ubiquity and
the great importance of clustering, we still do not have a satisfactory
mathematical theory of clustering. In order...
The complexity of a computational problem is traditionally quantified based
on the hardness of its worst case. This approach has many advantages and has
led to a deep and beautiful theory. However, from the practical perspective,
this leaves much to be desired. In application areas, practically interesting
instances very often occupy just a tiny pa...
We consider the following solitaire game whose rules are reminiscent of the
children's game of leapfrog. The player is handed an arbitrary ordering
$\pi=(x_1,x_2,...,x_n)$ of the elements of a finite poset $(P,\prec)$. At each
round an element may "skip over" the element in front of it, i.e. swap
positions with it. For example, if $x_i \prec x_{i+1...
Online reputation systems collect, maintain and disseminate reputations as a summary numerical score of past interactions of an establishment with its users. As reputation systems, including web search engines, gain in popularity and become a common method for people to select sought services, a dynamical system unfolds: Experts' reputation attract...
Fair allocation has been studied intensively in both economics and computer science. This subject has aroused renewed interest with the advent of virtualization and cloud computing. Prior work has typically focused on mechanisms for fair sharing of a single resource. We consider a variant where each user is entitled to a certain fraction of the sys...
We introduce oblivious protocols, a new framework for distributed computation with limited communication. Within this model we consider the musical chairs task MC(n,m), involving n players (processors) and m chairs. Initially, players occupy arbitrary chairs. Two players are in conflict if they both occupy the same chair. The task terminates when t...
We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19–35), in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O(λ ρ1/2). He...
This paper has two main focal points. We first consider an important class of machine learning algorithms - large margin classifiers, such as support vector machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative consta...
We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. This approach gives us access to several powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for...
We consider monotone embeddings of a finite metric space into low-dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into , while (in a sense to be made precise later), for alm...
In this paper we consider four previously known parameters of sign matrices from a complexity-theoretic perspective. The main technical contributions are tight (or nearly tight) inequalities that we establish among these parameters. Several new open problems are raised as well.
We consider monotone embeddings of a nite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l 2 , while, (in a sense to be made precise later), for...
Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n ew" eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range [ 2 d 1; 2 d 1] (If true, this i...
We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on $n$ points can be embedded into $l_2^n$, while, (in a sense to be made precise...
We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let $G$ be a graph on $n$ vertices. A 2-lift of $G$ is a graph $H$ on $2n$ vertices, with a covering map $\pi:H \to G$. It is not hard to see that all eigenvalues of $G$ are also eigenvalues of $H$...
We consider the problem of embedding a certain finite metric space to the Euclidean space, trying to keep the bi-Lipschitz constant as small as possible. We introduce the notationc
2(X, d) for the least distortion with which the metric space (X, d) may be embedded in a Euclidean space. It is known that if (X, d) is a metric space withn points, then...
Pairwise interaction models to recognize native folds are designed and analyzed. Different sets of parameters are considered but the focus was on 20 x 20 contact matrices. Simultaneous solution of inequalities and minimization of the variance of the energy find matrices that recognize exactly the native folds of 572 sequences and structures from th...
The ProtoMap site offers an exhaustive classification of all proteins in the SWISS-PROT database, into groups of related proteins.
The classification is based on analysis of all pairwise similarities among protein sequences. The analysis makes essential
use of transitivity to identify homologies among proteins. Within each group of the classificati...
We investigate the space of all protein sequences in search of clusters of related proteins. Our aim is to automatically detect these sets, and thus obtain a classification of all protein sequences. Our analysis, which uses standard measures of sequence similarity as applied to an all-vs.-all comparison of SWISSPROT, gives a very conservative initi...
We investigate the space of all protein sequences. We combine the standard measures of similarity (SW, FASTA, BLAST), to associate with each sequence an exhaustive list of neighboring sequences. These lists induce a (weighted directed) graph whose vertices are the sequences. The weight of an edge connecting two sequences represents their degree of...
A global classification of all currently known protein sequences is performed. Every protein sequence is partitioned into segments of 50 amino acid residues and a dynamic programming distance is calculated between each pair of segments. This space of segments is initially embedded into Euclidean space. The algorithm that we apply embeds every finit...
Given d, m and epsilon, we deterministically produce a sequence of points S that hits every combinatorial rectangle in [m]^d of volume at least epsilon. Both the running time of the algorithm and |S| are polynomial in m log(d)/epsilon. This algorithm has applications to deterministic constructions of small sample spaces for general multivalued rand...
In designing a routing scheme for a communication network it is desirable to use as short as possible paths for routing messages, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stretch factor - the maximum ratio between the cost of...
The distinct distribution of a binary code C is the sequence (B
<sub>i</sub>)<sub>i=0</sub><sup>n</sup> defined as follows: let B<sub>i
</sub>(w) be the number of codewords at distance i from the codeword w,
and let B<sub>i</sub> be the average of B<sub>i</sub>(w) over all w in
C. In this correspondence we study the distance distribution for codes...
In this paper we explore some implications of viewing graphs asgeometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect themetric of the (possibly weighted) graph. Gi...
The paper concerns some extremal problems on packing spheres in
graphs and covering graphs by spheres. Tight bounds are provided for
these problems on general graphs. The bounds are then applied to answer
the following question: Let f be a nonnegative function defined
on the vertices of a graph G , and suppose one has a lower bound
on the local ave...
The paper considers the computational hardness of approximating
the chromatic number of a graph. The authors first give a simple proof
that approximating the chromatic number of a graph to within a constant
power (of the value itself) in NP-hard. They then consider the hardness
of coloring a 3-colorable graph with as few as possible colors. They
sh...
We give an alternative proof to a characterization theorem of Gurvich for Boolean functions whose formula size is exactly the number of variables. These functions are called read-once functions. We use methods of combinatorial optimization and give, as a corollary, an alternative proof for some results of Seymour (1976, 1977).
Efficient two-party protocols for fault-tolerant computation of
any two-argument function are presented. It is proved that the influence
of a dishonest player in these protocols is the minimum one possible (up
to polylogarithmic factors). Also presented are efficient
m -party fault-tolerant protocols for sampling a general
distribution ( m ⩾2)....
We consider a simple abstract model for a class of discrete control processes, motivated in part by recent work about the behavior of imperfect random sources in computer algorithms. The process produces a string ofn bits and is a success or failure depending on whether the string produced belongs to a prespecified setL. In an uninfluenced process...
A radio network is a synchronous network of processors that communicate by transmitting messages to their neighbors. A processor receives a message in a given step if and only if it is silent then and precisely one of its neighbors transmits. This stringent rule poses serious difficulties in performing even the simplest tasks. This is true even und...
Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets
We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except fo...
In practice, almost all dynamic systems require decisions to be made online, without full knowledge of their future impact on the system. We introduce a general model for the processing of sequences of tasks and develop a general online decision algorithm. We show that, for an important class of special cases, this algorithm is optimal among all on...
We consider a simple model for a class of discrete control processes, motivated in part by recent work about the behavior of imperfect random sources in computer algorithms. The process produces a string of characters from {0, 1} of length n and is a “success” or “failure” depending on whether the string produced belongs to a prespecified set L. In...
Letf(n) (resp.g(n)) be the largestm such that there is a digraph (resp. a spanning weakly connected digraph) onn-vertices andm edges which is a subgraph of every tournament onn-vertices. We prove that
nlog2 n - c1 n \geqq f(n) \geqq g(n) \geqq nlog2 n - c2 nloglogn.n\log _2 n - c_1 n \geqq f(n) \geqq g(n) \geqq n\log _2 n - c_2 n\log \log n.