Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

Counting and Enumerating Crossing-free Geometric Graphs

We describe a framework for counting and enumerating various types of crossing-free geometric graphs on a planar point set. The framework generalizes ideas of Alvarez and Seidel, who used them to count triangulations in time $O(2^nn^2)$ where $n$ is the number of points. The main idea is to reduce the problem of counting geometric graphs to counting source-sink paths in a directed acyclic graph. The following new results will emerge. The number of all crossing-free geometric graphs can be computed in time $O(c^nn^4)$ for some $c < 2.83929$. The number of crossing-free convex partitions can be computed in time $O(2^nn^4)$. The number of crossing-free perfect matchings can be computed in time $O(2^nn^4)$. The number of convex subdivisions can be computed in time $O(2^nn^4)$. The number of crossing-free spanning trees can be computed in time $O(c^nn^4)$ for some $c < 7.04313$. The number of crossing-free spanning cycles can be computed in time $O(c^nn^4)$ for some $c < 5.61804$. With the same bounds on the running time we can construct data structures which allow fast enumeration of the respective classes. For example, after $O(2^nn^4)$ time of preprocessing we can enumerate the set of all crossing-free perfect matchings using polynomial time per enumerated object. For crossing-free perfect matchings and convex partitions we further obtain enumeration algorithms where the time delay for each (in particular, the first) output is bounded by a polynomial in $n$. All described algorithms are comparatively simple, both in terms of their analysis and implementation

Exponents and bounds for uniform spanning trees in d dimensions

Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such trees can be obtained in terms of the Laplacian matrix on the graph, by using Grassmann integrals. We use this to obtain exact exponents that bound those for the power-law decay of the probability that k distinct branches of the tree pass close to each of two distinct points, as the size of the lattice tends to infinity.Comment: 5 pages. v2: references added. v3: closed form results can be extended slightly (thanks to C. Tanguy). v4: revisions, and a figure adde

Output-Sensitive Tools for Range Searching in Higher Dimensions

Let $P$ be a set of $n$ points in ${\mathbb R}^{d}$. A point $p \in P$ is $k$\emph{-shallow} if it lies in a halfspace which contains at most $k$ points of $P$ (including $p$). We show that if all points of $P$ are $k$-shallow, then $P$ can be partitioned into $\Theta(n/k)$ subsets, so that any hyperplane crosses at most $O((n/k)^{1-1/(d-1)} \log^{2/(d-1)}(n/k))$ subsets. Given such a partition, we can apply the standard construction of a spanning tree with small crossing number within each subset, to obtain a spanning tree for the point set $P$, with crossing number $O(n^{1-1/(d-1)}k^{1/d(d-1)} \log^{2/(d-1)}(n/k))$. This allows us to extend the construction of Har-Peled and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set of $n$ points in ${\mathbb R}^{d}$ (without the shallowness assumption), a spanning tree $T$ with {\em small relative crossing number}. That is, any hyperplane which contains $w \leq n/2$ points of $P$ on one side, crosses $O(n^{1-1/(d-1)}w^{1/d(d-1)} \log^{2/(d-1)}(n/w))$ edges of $T$. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses $O(n \log \log n)$ space (and somewhat higher preprocessing cost), and answers a query in time $O(n^{1-1/(d-1)}k^{1/d(d-1)} (\log (n/k))^{O(1)})$, where $k$ is the output size

Fixed parameter tractability of crossing minimization of almost-trees

We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1-page book crossing number, the 2-page book crossing number, and the minimum number of crossed edges in 1-page and 2-page book drawings.Comment: Graph Drawing 201

Martin Gardner and His Influence on Recreational Math

Recreational mathematics is a relatively new field in the world of mathematics. While sometimes overlooked as frivolous since those who practice it need no advanced knowledge of the subject, recreational mathematics is a perfect transition for people to experience the joy in logically establishing a solution. Martin Gardner recognized that this pattern of proving solutions to questions is how mathematics progresses. From his childhood on, Gardner greatly influenced the mathematical world. Although not a mathematician, he inspired many to pursue careers and make advancements in mathematics during his 25-year career with Scientific American. He encouraged novices to expand their knowledge, enlightened professionals of computer science developments, and established his own proofs