arXiv subjects

Noga Alon

Publications and source records attributed to Noga Alon.

At least 145 records · Page 8Linked to original sources

arXiv2015

Local And Global Colorability of Graphs

It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tildeΘ\left(n ^ {\frac{1}{r+1}} \right)$ (that is, $n^\frac{1}{r+1}$ up to a factor poly-logarithmic in $n$). The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random $n$-vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius $r$ in it are $2$-degenerate.

math.CO↗

arXiv2015

Coloring, sparseness, and girth

An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-$k$-choosable bipartite graphs, showing that maximum average degree at most $2(k-1)$ is a sharp sufficient condition for $k$-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-$k$-colorable graphs and hypergraphs with any girth $g$.

math.CO↗

arXiv2015

Welfare Maximization with Limited Interaction

We continue the study of welfare maximization in unit-demand (matching) markets, in a distributed information model where agent's valuations are unknown to the central planner, and therefore communication is required to determine an efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed that if the market size is $n$, then $r$ rounds of interaction (with logarithmic bandwidth) suffice to obtain an $n^{1/(r+1)}$-approximation to the optimal social welfare. In particular, this implies that such markets converge to a stable state (constant approximation) in time logarithmic in the market size. We obtain the first multi-round lower bound for this setup. We show that even if the allowable per-round bandwidth of each agent is $n^{ε(r)}$, the approximation ratio of any $r$-round (randomized) protocol is no better than $Ω(n^{1/5^{r+1}})$, implying an $Ω(\log \log n)$ lower bound on the rate of convergence of the market to equilibrium. Our construction and technique may be of interest to round-communication tradeoffs in the more general setting of combinatorial auctions, for which the only known lower bound is for simultaneous ($r=1$) protocols [DNO14].

cs.GT↗

arXiv2015

Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with $\widetildeΘ(α^{1/2} T^{1/2})$ minimax regret, where $α$ is the independence number of the underlying graph; the second class induces problems with $\widetildeΘ(δ^{1/3}T^{2/3})$ minimax regret, where $δ$ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time.

cs.LG↗

arXiv2014

Splitting necklaces and measurable colorings of the real line

A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split by at most $k$ cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every $k\geq 1$ there is a measurable $(k+3)$-coloring of the real line such that no interval can be fairly split using at most $k$ cuts. In particular, there is a measurable $4$-coloring of the real line in which no two adjacent intervals have the same measure of every color. An analogous problem for the integers was posed by Erdős in 1961 and solved in the affirmative in 1991 by Keränen. Curiously, in the discrete case the desired coloring also uses four colors.

math.CO↗

arXiv2014

Comparable pairs in families of sets

Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. In this paper we study a complementary problem posed by Erdős and Daykin and Frankl in the early '80s. They asked for the maximum number of comparable pairs that can appear in a family of $m$ subsets of $[n]$, a quantity we denote by $c(n,m)$. We first resolve an old conjecture of Alon and Frankl, showing that $c(n,m) = o(m^2)$ when $m = n^{ω(1)} 2^{n/2}$. We also obtain more accurate bounds for $c(n,m)$ for sparse and dense families, characterise the extremal constructions for certain values of $m$, and sharpen some other known results.

math.CO↗

arXiv2014

Nonstochastic Multi-Armed Bandits with Graph-Structured Feedback

We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide several algorithms addressing different variants of our setting, and provide tight regret bounds depending on combinatorial properties of the information feedback structure.

cs.LG↗

arXiv2014

More on the bipartite decomposition of random graphs

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n -α(G)$, where $α(G)$ is the maximum size of an independent set of $G$. Erdős conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $bc(G)=n-α(G)$ with high probability, that is, with probability that tends to 1 as $n$ tends to infinity. The first author showed that this is slightly false, proving that for most values of $n$ tending to infinity and for $G=G(n,0.5)$, $bc(G) \leq n-α(G)-1$ with high probability. We prove a stronger bound: there exists an absolute constant $c>0$ so that $bc(G) \leq n-(1+c)α(G)$ with high probability.

math.CO↗

arXiv2014

Broadcast Throughput in Radio Networks: Routing vs. Network Coding

The broadcast throughput in a network is defined as the average number of messages that can be transmitted per unit time from a given source to all other nodes when time goes to infinity. Classical broadcast algorithms treat messages as atomic tokens and route them from the source to the receivers by making intermediate nodes store and forward messages. The more recent network coding approach, in contrast, prompts intermediate nodes to mix and code together messages. It has been shown that certain wired networks have an asymptotic network coding gap, that is, they have asymptotically higher broadcast throughput when using network coding compared to routing. Whether such a gap exists for wireless networks has been an open question of great interest. We approach this question by studying the broadcast throughput of the radio network model which has been a standard mathematical model to study wireless communication. We show that there is a family of radio networks with a tight $Θ(\log \log n)$ network coding gap, that is, networks in which the asymptotic throughput achievable via routing messages is a $Θ(\log \log n)$ factor smaller than that of the optimal network coding algorithm. We also provide new tight upper and lower bounds that show that the asymptotic worst-case broadcast throughput over all networks with $n$ nodes is $Θ(1 / \log n)$ messages-per-round for both routing and network coding.

cs.DS↗

arXiv2014

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $Θ(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.

math.CO↗

arXiv2014

Chasing robbers on random geometric graphs---an alternative approach

We study the vertex pursuit game of \emph{Cops and Robbers}, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. We focus on $G_{d}(n,r)$, a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^d$, and two vertices are adjacent if the Euclidean distance between them is at most $r$. The main result is that if $r^{3d-1}>c_d \frac{\log n}{n}$ then the cop number is $1$ with probability that tends to $1$ as $n$ tends to infinity. The case $d=2$ was proved earlier and independently in \cite{bdfm}, using a different approach. Our method provides a tight $O(1/r^2)$ upper bound for the number of rounds needed to catch the robber.

math.CO↗

arXiv2014

The Turán number of sparse spanning graphs

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $δ(H)>0$ and $Δ(H)$ denote the minimum degree and maximum degree of $H$, respectively. We prove that for all $n$ sufficiently large, if $H$ is any graph of order $n$ with $Δ(H) \le \sqrt{n}/200$, then $ex(n,H)={{n-1} \choose 2}+δ(H)-1$. The condition on the maximum degree is tight up to a constant factor. This generalizes a classical result of Ore for the case $H=C_n$, and resolves, in a strong form, a conjecture of Glebov, Person, and Weps for the case of graphs. A counter-example to their more general conjecture concerning the extremal number of bounded degree spanning hypergraphs is also given.

math.CO↗

arXiv2014

Bipartite decomposition of random graphs

For a graph $G=(V,E)$, let $τ(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $τ(G) \leq n -α(G)$, where $α(G)$ is the maximum size of an independent set of $G$. Erdős conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $τ(G)=n-α(G)$ with high probability, that is, with probability that tends to $1$ as $n$ tends to infinity. Here we show that this conjecture is (slightly) false, proving that for most values of $n$ tending to infinity and for $G=G(n,0.5)$, $τ(G) \leq n-α(G)-1$ with high probability, and that for some sequences of values of $n$ tending to infinity $τ(G) \leq n-α(G)-2$ with probability bounded away from $0$. We also study the typical value of $τ(G)$ for random graphs $G=G(n,p)$ with $p < 0.5$ and show that there is an absolute positive constant $c$ so that for all $p \leq c$ and for $G=G(n,p)$, $τ(G)=n-Θ(α(G))$ with high probability.

math.CO↗

arXiv2014

Two notions of unit distance graphs

A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly $1$. A {\em (unit) distance graph} in $\mathbb{R}^d$ is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed $d$ the number of faithful distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1+o(1)) d n \log_2 n}$, and give a short proof of the known fact that the number of distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1-1/\lfloor d/2 \rfloor +o(1))n^2/2}$. We also study the behavior of several Ramsey-type quantities involving these graphs. % and high-girth graphs from these classes. In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in $\mathbb R^d$.

math.CO↗

arXiv2014

A note on general sliding window processes

Let $f:\mathbb{R}^k\to \mathbb{R}$ be a measurable function, and let $\{U_i\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables. Consider the random process $Z_i=f(U_{i},...,U_{i+k-1})$. We show that for all $\ell$, there is a positive probability, uniform in $f$, for $Z_1,...,Z_\ell$ to be monotone. We give upper and lower bounds for this probability, and draw corollaries for $k$-block factor processes with a finite range. The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.

math.PR↗

arXiv2014

Maximizing the number of nonnegative subsets

Given a set of $n$ real numbers, if the sum of elements of every subset of size larger than $k$ is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is $\binom{n-1}{k-1} + \binom{n-1}{k-2} + \cdots + \binom{n-1}{0}+1$, settling a problem of Tsukerman. We provide two proofs, the first establishes and applies a weighted version of Hall's Theorem and the second is based on an extension of the nonuniform Erdős-Ko-Rado Theorem.

math.CO↗

arXiv2013

From Bandits to Experts: A Tale of Domination and Independence

We consider the partial observability model for multi-armed bandits, introduced by Mannor and Shamir. Our main result is a characterization of regret in the directed observability model in terms of the dominating and independence numbers of the observability graph. We also show that in the undirected case, the learner can achieve optimal regret without even accessing the observability graph before selecting an action. Both results are shown using variants of the Exp3 algorithm operating on the observability graph in a time-efficient manner.

cs.LG↗

arXiv2012

Local Correction with Constant Error Rate

A Boolean function f of n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma(x)=f_sigma(x_1, ..., x_n) = f(x_sigma(1), ..., x_sigma(n)) of f, we can compute f_sigma(x) for any x in {0,1}^n with good probability using q queries to g. It is known that degree d polynomials are O(2^d)-locally correctable, and that most k-juntas are O(k log k)-locally correctable, where the closeness parameter, or more precisely the distance between g and f_sigma, is required to be exponentially small (in d and k respectively). In this work we relax the requirement for the closeness parameter by allowing the distance between the functions to be a constant. We first investigate the family of juntas, and show that almost every k-junta is O(k log^2 k)-locally correctable for any distance epsilon < 0.001. A similar result is shown for the family of partially symmetric functions, that is functions which are indifferent to any reordering of all but a constant number of their variables. For both families, the algorithms provided here use non-adaptive queries and are applicable to most but not all functions of each family (as it is shown to be impossible to locally correct all of them). Our approach utilizes the measure of symmetric influence introduced in the recent analysis of testing partial symmetry of functions.

cs.CC↗