arXiv subjects

Daniel Reichman

Publications and source records attributed to Daniel Reichman.

41 records · Page 3Linked to original sources

arXiv2014

Contagious Sets in Expanders

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors, where $r>1$ is the activation threshold. A \emph{contagious set} is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set. Computing $m(G,r)$ is NP-hard. It is known that for every $d$-regular or nearly $d$-regular graph on $n$ vertices, $m(G,r) \le O(\frac{nr}{d})$. We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion (e.g., $λ(G)=O(\sqrt{d})$, or girth $Ω(\log \log d)$) implies that $m(G,2) \le O(\frac{n}{d^2})$ (and more generally, $m(G,r) \le O(\frac{n}{d^{r/(r-1)}})$). Significantly weaker expansion properties suffice in order to imply that $m(G,2)\le O(\frac{n \log d}{d^2})$. For example, we show this for graphs of girth at least~7, and for graphs with $λ(G)<(1-ε)d$, provided the graph has no 4-cycles. Nearly $d$-regular expander graphs can be obtained by considering the binomial random graph $G(n,p)$ with $p \simeq \frac{d}{n}$ and $d > \log n$. For such graphs we prove that $Ω(\frac{n}{d^2 \log d}) \le m(G,2) \le O(\frac{n\log\log d}{d^2\log d})$ almost surely. Our results are algorithmic, entailing simple and efficient algorithms for selecting contagious sets.

cs.DM↗

arXiv2014

On giant components and treewidth in the layers model

Given an undirected $n$-vertex graph $G(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex induced subgraph of $G$ generated by ordering $V$ according to a random permutation $π$ and including in $T_k(G)$ those vertices with at most $k-1$ of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} $k$. The layers model has found applications in studying $\ell$-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are $3$-regular graphs $G$ for which with high probability $T_3(G)$ has a connected component of size $Ω(n)$. Moreover, this connected component has treewidth $Ω(n)$. This lower bound on the treewidth extends to many other random graph models. In contrast, $T_2(G)$ is known to be a forest (hence of treewidth~1), and we establish that if $G$ is of bounded degree then with high probability the largest connected component in $T_2(G)$ is of size $O(\log n)$. We also consider the infinite two-dimensional grid, for which we prove that the first four layers contain a unique infinite connected component with probability $1$.

cs.DM↗

arXiv2011

Tight Approximation of Image Matching

In this work we consider the {\em image matching} problem for two grayscale $n \times n$ images, $M_1$ and $M_2$ (where pixel values range from 0 to 1). Our goal is to find an affine transformation $T$ that maps pixels from $M_1$ to pixels in $M_2$ so that the differences over pixels $p$ between $M_1(p)$ and $M_2(T(p))$ is minimized. Our focus here is on sublinear algorithms that give an approximate result for this problem, that is, we wish to perform this task while querying as few pixels from both images as possible, and give a transformation that comes close to minimizing the difference. We give an algorithm for the image matching problem that returns a transformation $T$ which minimizes the sum of differences (normalized by $n^2$) up to an additive error of $ε$ and performs $\tilde{O}(n/ε^2)$ queries. We give a corresponding lower bound of $Ω(n)$ queries showing that this is the best possible result in the general case (with respect to $n$ and up to low order terms). In addition, we give a significantly better algorithm for a natural family of images, namely, smooth images. We consider an image smooth when the total difference between neighboring pixels is O(n). For such images we provide an approximation of the distance between the images to within an additive error of $ε$ using a number of queries depending polynomially on $1/ε$ and not on $n$. To do this we first consider the image matching problem for 2 and 3-dimensional {\em binary} images, and then reduce the grayscale image matching problem to the 3-dimensional binary case.

cs.DS↗

arXiv2011

Recoverable Values for Independent Sets

The notion of {\em recoverable value} was advocated in work of Feige, Immorlica, Mirrokni and Nazerzadeh [Approx 2009] as a measure of quality for approximation algorithms. There this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has {\em recoverable value} $ρ$, if for every graph it recovers an independent set of size at least $\max_I \sum_{v\in I} \min[1,ρ/(d(v) + 1)]$, where $d(v)$ is the degree of vertex $v$, and $I$ ranges over all independent sets in $G$. Hence, in a sense, from every vertex $v$ in the maximum independent set the algorithm recovers a value of at least $ρ/(d_v + 1)$ towards the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. It easily follows from known results that some simple algorithms for MIS ensure $ρ\ge 1$. We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least $ρ\ge 7/3$. In addition, we show that approximating MIS in graphs with a given $k$-coloring within a ratio larger than $2/k$ is unique games hard. This rules out a natural approach for obtaining $ρ\ge 2$.

cs.DS↗

arXiv2010

Influence is a Matter of Degree: New Algorithms for Activation Problems

We consider the target set selection problem. In this problem, a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $k$ active neighbors ($k$ is identical for all vertices of the graph). Our goal is to find a set of minimum size whose activation will result with the entire graph being activated. Call such a set \emph{contagious}. We prove that if $G=(V,E)$ is an undirected graph, the size of a contagious set is bounded by $\sum_{v\in V}{\min \{1,\frac{k}{d(v)+1}\}}$ (where $d(v)$ is the degree of $v$). We present a simple and efficient algorithm that finds a contagious set that is not larger than the aforementioned bound and discuss algorithmic applications of this algorithm to finding contagious sets in dense graphs.

cs.DM↗