The complexity of separating points in the plane

We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles

Minimum Cell Connection in Line Segment Arrangements

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points a and b in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting a to b that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting a to b must stay inside a given polygon P with a constant number of holes, the segments are contained in P, and the endpoints of the segments are on the boundary of P. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution

Minimum cell connection in line segment arrangements

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points a and b in different cells of the induced arrangement: (i) compute the minimum number of segments one needs to remove so that there is a path connecting a to b that does not intersect any of the remaining segments; (ii) compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting a to b must stay inside a given polygon P with a constant number of holes, the segments are contained in P, and the endpoints of the segments are on the boundary of P. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution

Estimation in high dimensions: a geometric perspective

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change

The Complexity of Separating Points in the Plane

We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles

The parameterized complexity of some geometric problems in unbounded dimension

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension $d$: i) Given $n$ points in \Rd, compute their minimum enclosing cylinder. ii) Given two $n$-point sets in \Rd, decide whether they can be separated by two hyperplanes. iii) Given a system of $n$ linear inequalities with $d$ variables, find a maximum-size feasible subsystem. We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension $d$. %and hence not solvable in ${O}(f(d)n^c)$ time, for any computable function $f$ and constant $c$ %(unless FPT=W[1]). Our reductions also give a $n^{\Omega(d)}$-time lower bound (under the Exponential Time Hypothesis)

Enhanced neural response to anticipation, effort and consummation of reward and aversion during Bupropion treatment

Background We have previously shown that the selective serotonergic re-uptake inhibitor, citalopram, reduces the neural response to reward and aversion in healthy volunteers. We suggest that this inhibitory effect might underlie the emotional blunting reported by patients on these medications. Bupropion is a dopaminergic and noradrenergic re-uptake inhibitor and has been suggested to have more therapeutic effects on reward-related deficits. However, how bupropion affects the neural responses to reward and aversion is unclear. Methods 17 healthy volunteers (9 female, 8 male) received 7 days of bupropion (150 mg/day) and 7 days of placebo treatment, in a double-blind crossover design. Our functional Magnetic Resonance Imaging task consisted of 3 phases; an anticipatory phase (pleasant or unpleasant cue), an effort phase (button presses to achieve a pleasant taste or to avoid an unpleasant taste) and a consummatory phase (pleasant or unpleasant tastes). Volunteers also rated wanting, pleasantness and intensity of the tastes. Results Relative to placebo, bupropion increased activity during the anticipation phase in the ventral medial prefrontal cortex (vmPFC) and caudate. During the effort phase, bupropion increased activity in the vmPFC, striatum, dorsal anterior cingulate cortex and primary motor cortex. Bupropion also increased medial orbitofrontal cortex, amygdala and ventral striatum activity during the consummatory phase. Conclusions Our results are the first to show that bupropion can increase neural responses during the anticipation, effort and consummation of rewarding and aversive stimuli. This supports the notion that bupropion might be beneficial for depressed patients with reward-related deficits and blunted affect

On k-means for segments and polylines

We study the problem of k-means clustering in the space of straight-line segments in R² under the Hausdorff distance. For this problem, we give a (1 + ε)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n + ε−O(k) + ε−O(k)· logO(k)(ε−1)). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron [40] to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg [21, 22]. Our results can be extended to polylines of constant complexity with a runningtime of O(n + ε−O(k))

On kk-means for segments and polylines

We study the problem of k-means clustering in the space of straight-line segments in R2 under the Hausdorff distance. For this problem, we give a (1+ϵ)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n+ϵ−O(k)+ϵ−O(k)⋅logO(k)(ϵ−1)). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron~\cite{Vigneron14} to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg~\cite{Feldman11, Feldman2020}. Our results can be extended to polylines of constant complexity with a running time of O(n+ϵ−O(k))