Single-Player and Two-Player Buttons & Scissors Games

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.Comment: 21 pages, 15 figures. Presented at JCDCG2 2015, Kyoto University, Kyoto, Japan, September 14 - 16, 201

Packing plane spanning trees and paths in complete geometric graphs

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph GKn on any set S of n points in general position in the plane? We show that this number is in Ω(n). Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).</p

Approximating Largest Convex Hulls for Imprecise Points

Assume that a set of imprecise points in the plane is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NP-hardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a variety of models, based on the core-set paradigm

Automated Puzzle Difficulty Estimation

We introduce a method for automatically rating the difficulty of puzzle game levels. Our method takes multiple aspects of the levels of these games, such as level size, and combines these into a difficulty function. It can simply be adapted to most puzzle games, and we test it on three different ones: Flow, Lazors and Move. We conducted a user study to discover how difficult players find the levels of a set and use this data to train the difficulty function to match the user-provided ratings. Our experiments show that the difficulty function is capable of rating levels with an average error of approximately one point in Lazors and Move, and less than half a point in Flow, on a difficulty scale of 1-10

Computing Morphological Properties of Arrangements of Lines

An arrangement of n lines in the plane is a partition of the plane into O(n²) faces, edges, and vertices (intersection points). Such line processes play a fundamental role in modeling spatial patterns and studying a variety of problems such as traffic flow. We briefly survey recent results on the complexity of computing morphological properties of such arrangements

Mixed Map Labeling

Point feature map labeling is a geometric problem, in which a set of input points must be labeled with a set of disjoint rectangles (the bounding boxes of the label texts). Typically, labeling models either use internal labels, which must touch their feature point, or external (boundary) labels, which are placed on one of the four sides of the input points’ bounding box and which are connected to their feature points by crossing-free leader lines. In this paper we study polynomial-time algorithms for maximizing the number of internal labels in a mixed labeling model that combines internal and external labels. The model requires that all leaders are parallel to a given orientation θ∈[0,2π) , whose value influences the geometric properties and hence the running times of our algorithms