Daniel Reichman

FasT-Match: Fast Affine Template Matching

Fast-Match is a fast algorithm for approximate template matching under 2D affine transformations that minimizes the Sum-of-Absolute-Differences (SAD) error measure. There is a huge number of transformations to consider but we prove that they can be sampled using a density that depends on the smoothness of the image. For each potential transformation, we approximate the SAD error using a sub linear algorithm that randomly examines only a small number of pixels. We further accelerate the algorithm using a branch-and-bound scheme. As images are known to be piecewise smooth, the result is a practical affine template matching algorithm with approximation guarantees, that takes a few seconds to run on a standard machine. We perform several experiments on three different datasets, and report very good results. To the best of our knowledge, this is the first template matching algorithm which is guaranteed to handle arbitrary 2D affine transformations.

On Systems of Linear Equations with Two Variables per Equation

For a prime p, max-2linp is the problem of satisfying as many equations as possible from a system of linear equations modulo p, where every equation contains two variables. Hastad shows that this problem is NP-hard to approximate within a ratio of 11/12 + e for p=2, and Andersson, Engebretsen and Hastad show the same hardness of approximation ratio for p ≥ 11, and somewhat weaker results (such as 69/70) for p = 3,5,7. We prove that max-2linp is easiest to approximate when p = 2, implying for every prime p that max-2linp is NP-hard to approximate within a ratio of 11/12 + e. For large p, we prove stronger hardness of approximation results. Namely, we show that there is some universal constant δ > 0 such that it is NP-hard to approximate max-2linp within a ratio better than 1/p δ . We use our results so as to clarify some aspects of Khot’s unique games conjecture. Namely, we show that for every e > 0 it is NP-hard to approximate the value of unique games within a ratio of e.

Contagious Sets in Dense Graphs

We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m(G,r) be the size of a smallest contagious set in a graph G on n vertices. We examine density conditions that ensure m(G,r) = r for all r >= 2. With respect to the minimum degree, we prove that such a smallest possible contagious set is guaranteed to exist if and only if G has minimum degree at least (k-1)/k * n. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such a graph. We also investigate what average degree asserts the existence of small contagious sets. For n >= k >= r, we denote by M(n,k,r) the maximum number of edges in an n-vertex graph G satisfying m(G,r)>k. We determine the precise value of M(n,k,2) and M(n,k,k), assuming that n is sufficiently large compared to k.

On the hardness of approximating Max-Satisfy

Max-Satisfy is the problem of finding an assignment that satisfies the maximum number of equations in a system of linear equations over Q. We prove that unless NP ⊂ BPP Max-Satisfy cannot be efficiently approximated within an approximation ratio of 1/n1-e, if we consider systems of n linear equations with at most n variables and e > 0 is an arbitrarily small constant. Previously, it was known that the problem is NP-hard to approximate within a ratio of 1/nα, but 0 < α < 1 was some specific constant that could not be taken to be arbitrarily close to 1.

Contagious sets in random graphs

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

Fast-Match: Fast Affine Template Matching

r

Smoothed analysis on connected graphs.

active neighbors. A \emph{contagious set} is a set whose activation results with the entire graph being active. Given a graph

Recoverable values for independent sets

G

Contagious sets in dense graphs

, let

Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

m(G,r)