arXiv subjects

Nati Linial

Publications and source records attributed to Nati Linial.

At least 55 records · Page 3Linked to original sources

arXiv2015

On the number of 4-cycles in a tournament

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 random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of $4$-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in $T$, how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.

math.CO↗

arXiv2015

On the Rigidity of Sparse Random Graphs

A graph with a trivial automorphism group is said to be rigid. Wright proved that for $\frac{\log n}{n}+ω(\frac 1n)\leq p\leq \frac 12$ a random graph $G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is sharp and for $p<\frac{(1-ε)\log n}{n}$ with positive probability $\text{aut}(G)$ is nontrivial. We show that in the sparser case $ω(\frac 1 n)\leq p\leq \frac{\log n}{n}+ω(\frac 1n)$, it holds whp that $G$'s $2$-core is rigid. We conclude that for all $p$, a graph in $G(n,p)$ is reconstrutible whp. In addition this yields for $ω(\frac 1n)\leq p\leq \frac 12$ a canonical labeling algorithm that almost surely runs in polynomial time with $o(1)$ error rate. This extends the range for which such an algorithm is currently known.

math.CO↗

arXiv2014

On the local profiles of trees

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.

math.CO↗

arXiv2014

A Note on the Inducibility of 4-vertex Graphs

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 addressed in a construction of Exoo (1986), but remains unknown. Refuting a conjecture of Erdos, Thomason (1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

math.CO↗

arXiv2014

Market Share Indicates Quality

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 fixed-price model where customers are informed about their history with products and about market share data. We find that it is in fact correct to make a Bayesian inference that the product with the higher market share has the better quality under few and unrestrictive assumptions on customer behavior.

cs.GT↗

arXiv2014

The complexity of learning halfspaces using generalized linear methods

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 addition they are widely used in numerous application domains. Despite their importance, there are still very few proof techniques to show limits on the power of these algorithms. We study the performance of this approach in the problem of (agnostically and improperly) learning halfspaces with margin $γ$. Let $\mathcal{D}$ be a distribution over labeled examples. The $γ$-margin error of a hyperplane $h$ is the probability of an example to fall on the wrong side of $h$ or at a distance $\leγ$ from it. The $γ$-margin error of the best $h$ is denoted $\mathrm{Err}_γ(\mathcal{D})$. An $α(γ)$-approximation algorithm receives $γ,ε$ as input and, using i.i.d. samples of $\mathcal{D}$, outputs a classifier with error rate $\le α(γ)\mathrm{Err}_γ(\mathcal{D}) + ε$. Such an algorithm is efficient if it uses $\mathrm{poly}(\frac{1}γ,\frac{1}ε)$ samples and runs in time polynomial in the sample size. The best approximation ratio achievable by an efficient algorithm is $O\left(\frac{1/γ}{\sqrt{\log(1/γ)}}\right)$ and is achieved using an algorithm from the above class. Our main result shows that the approximation ratio of every efficient algorithm from this family must be $\ge Ω\left(\frac{1/γ}{\mathrm{poly}\left(\log\left(1/γ\right)\right)}\right)$, essentially matching the best known upper bound.

cs.LG↗

arXiv2014

Triply Existentially Complete Triangle-Free Graphs

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 3-existentially complete triangle-free graphs.

math.CO↗

arXiv2014

From average case complexity to improper learning complexity

The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing lower bounds fall short of the best known algorithms. The biggest challenge in proving complexity results is to establish hardness of {\em improper learning} (a.k.a. representation independent learning).The difficulty in proving lower bounds for improper learning is that the standard reductions from $\mathbf{NP}$-hard problems do not seem to apply in this context. There is essentially only one known approach to proving lower bounds on improper learning. It was initiated in (Kearns and Valiant 89) and relies on cryptographic assumptions. We introduce a new technique for proving hardness of improper learning, based on reductions from problems that are hard on average. We put forward a (fairly strong) generalization of Feige's assumption (Feige 02) about the complexity of refuting random constraint satisfaction problems. Combining this assumption with our new technique yields far reaching implications. In particular, 1. Learning $\mathrm{DNF}$'s is hard. 2. Agnostically learning halfspaces with a constant approximation ratio is hard. 3. Learning an intersection of $ω(1)$ halfspaces is hard.

cs.LG↗

arXiv2014

Graphs with few 3-cliques and 3-anticliques are 3-universal

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.

math.CO↗

arXiv2013

On the 3-local profiles of graphs

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 description of the projection of ${\cal S}_3$ to the (p_0, p_3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p_0+p_3\geq 1/4. We also give a full description of the triangle-free case, i.e., the intersection of ${\cal S}_3$ with the hyperplane p_3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

math.CO↗

arXiv2013

On the densities of cliques and independent sets in graphs

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 theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.

math.CO↗

arXiv2013

More data speeds up training time in learning halfspaces over sparse vectors

The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a {\em computational} resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a {\em natural supervised learning problem} --- we consider agnostic PAC learning of halfspaces over $3$-sparse vectors in $\{-1,1,0\}^n$. This class is inefficiently learnable using $O\left(n/ε^2\right)$ examples. Our main contribution is a novel, non-cryptographic, methodology for establishing computational-statistical gaps, which allows us to show that, under a widely believed assumption that refuting random $\mathrm{3CNF}$ formulas is hard, it is impossible to efficiently learn this class using only $O\left(n/ε^2\right)$ examples. We further show that under stronger hardness assumptions, even $O\left(n^{1.499}/ε^2\right)$ examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using $\tildeΩ\left(n^2/ε^2\right)$ examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.

cs.LG↗

arXiv2013

Internal Partitions of Regular Graphs

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 leads to interesting new open problems on cubic graphs. We also provide new lower bounds on $N(d)$ and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well.

math.CO↗

arXiv2013

On high-dimensional acyclic tournaments

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 $Ω(\log^{1/d}n)$ vertices and the bound is tight. This statement for tournaments (i.e., the case $d=1$) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the inter-relations among various other notions of acyclicity in high-dimensional to tournaments. These include combinatorial, geometric and topological concepts.

math.CO↗

arXiv2013

The threshold for collapsibility in random complexes

In this paper we determine the threshold for collapsibility in the probabilistic model $X_d(n,p)$ of $d$-dimensional simplicial complexes. A lower bound for this threshold $p=\frac{c_d}{n}$ was established in \cite{ALLM}. Here we show that this is indeed the correct threshold. Namely, for every $c>c_d$, a complex drawn from $X_d(n,\frac{c}{n})$ is asymptotically almost surely not collapsible.

math.PR↗

arXiv2012

Musical chairs

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 (and a win of the team) is declared as soon as each player occupies a unique chair. Two players that simultaneously occupy the same chair are said to be {\em in conflict}. In other words, termination (and a win for the team) is reached as soon as there are no conflicts. The only means of communication throughout the game is this: At every round of the game, the scheduler selects an arbitrary nonempty set of players who are currently in conflict, and notifies each of them separately that it must move. A player who is thus notified changes its chair according to its deterministic program. As we show, for $m\ge 2n-1$ chairs the team has a winning strategy. Moreover, using topological arguments we show that this bound is tight. For $m\leq 2n-2$ the scheduler has a strategy that is guaranteed to make the game continue indefinitely and thus win. We also have some results on additional interesting questions. For example, if $m \ge 2n-1$ (so that the team can win), how quickly can they achieve victory?

math.CO↗

arXiv2012

Clustering is difficult only when it does not matter

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 to properly understand clustering, it is clearly necessary to develop a solid theoretical basis for the area. For example, from the perspective of computational complexity theory the clustering problem seems very hard. Numerous papers introduce various criteria and numerical measures to quantify the quality of a given clustering. The resulting conclusions are pessimistic, since it is computationally difficult to find an optimal clustering of a given data set, if we go by any of these popular criteria. In contrast, the practitioners' perspective is much more optimistic. Our explanation for this disparity of opinions is that complexity theory concentrates on the worst case, whereas in reality we only care for data sets that can be clustered well. We introduce a theoretical framework of clustering in metric spaces that revolves around a notion of "good clustering". We show that if a good clustering exists, then in many cases it can be efficiently found. Our conclusion is that contrary to popular belief, clustering should not be considered a hard task.

cs.LG↗

arXiv2012

On the practically interesting instances of MAXCUT

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 part of an algorithm's space of instances, and the vast majority of instances are simply irrelevant. Addressing these issues is a major challenge for theoretical computer science which may make theory more relevant to the practice of computer science. Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, $(1+ε)$-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time $Ω(\sqrt{n})$-stable instances of MAXCUT, substantially improving the best previously known result.

cs.CC↗