Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension

We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta)

An approximation algorithm for the art gallery problem

Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter

Parameterized Hardness of Art Gallery Problems

International audienceGiven a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line 2 segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S. The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out any f (k)n o(k /log k) algorithm, where k := |S | is the number of guards, for any computable function f , unless the Exponential Time Hypothesis fails. These lower bounds almost match the n O (k) algorithms that exist for both problems

Model-based Stochastical Segmentation of Higher-dimensional Data

This thesis is motivated by the problem of segmenting extremely noisy images of geometric objects. To this end, it combines randomized combinatorial set cover optimization with a statistical model of object interaction. The set cover approach provides stability and applicability in cases in which many traditional methods of segmentation fail due to noise and imperfect data. The statistical model provides additional information that is not directly supplied by the image, and leads to a more realistic depiction of physical object properties in the resulting segmentation. This dissertation is divided into three parts: The first covers topics of randomized combinatorial optimization. This includes improving bounds of convergence and establishing a method of parallelization for an existing approach, as well as linking solutions to different combinatorial problems, such as geometric set cover and a general linear program. Part two is concerned with constructing a point process model of object interaction that fits later applications, and exploring some theoretical and practical pitfalls in its simulation, estimation, and coupling with a combinatorial approach. Part three compares previously discussed methods empirically, and demonstrates the performance of the established combination of randomized optimization and statistical model on microscopic cell images and 3D μCT scans of fiber reinforced materials