Computational Geometry-Theory and Applications最新文献

In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that $(1+\epsilon )$-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have only a linear dependence on n.

Let $t>1$ be a real number. A geometric t-spanner is a geometric graph for a point set in $R^d$ with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t.

An imprecise point set is modeled by a set R of regions in $R^d$. If one chooses a point inside each region of R, then the resulting point set is called a precise instance from R. An imprecise t-spanner for an imprecise point set R is a graph $G=(R,E)$ such that for each precise instance S from R, graph $G_S=(S,E_S)$, where $E_S$ is the set of edges corresponding to E and S, is a t-spanner.

In this paper, we show an imprecise point set R of n straight-line segments in the plane such that any imprecise t-spanner for R has $\Omega \left(n^2\right)$ edges. Then, we give an algorithm that computes an imprecise t-spanner for a set of n pairwise disjoint d-dimensional balls with arbitrary sizes. This imprecise t-spanner has $O(n/{(t-1)}^d)$ edges and can be computed in $O(n\log n/{(t-1)}^d)$ time. Finally, we show that given an imprecise spanner, finding a precise instance such that its corresponding precise spanner has minimum dilation between all possible precise instances of the imprecise spanner is NP-hard, no matter if crossing edges are allowed or not.

We study the problem of augmenting a metric graph by adding k edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time $(5/3-ϵ)$-approximation algorithm, for any $ϵ>0$, unless $P=NP$.

We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs with bounded treewidth.

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

We show that several classes of polyhedra are joined by a sequence of $O\left(1\right)$ refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain a ≤6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron.

In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominating-set problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the set-cover problem.

Both problems have been well-studied, subject to various restrictions on the input objects. These problems are $\mathrm{APX}$-hard for object sets consisting of axis-parallel rectangles, ellipses, α-fat objects of constant description complexity, and convex polygons. On the other hand, $\mathrm{PTAS}$s (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a $\mathrm{PTAS}$ was unknown even for arbitrary squares. For both problems obtaining a $\mathrm{PTAS}$ remains open for a large class of objects.

For the dominating-set problem, we prove that a popular local-search algorithm leads to a $(1+\epsilon )$ approximation for a family of homothets of a convex object (which includes arbitrary squares, k-regular polygons, translated and scaled copies of a convex set, etc.) in $n^{O(1/{\epsilon }^2)}$ time. On the other hand, the same approach leads to a $\mathrm{PTAS}$ for the geometric covering problem when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easy-to-implement approximation algorithm for both problems for a large class of objects, significantly improving the best-known approximation guarantees.

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges such that edges of the same color do not cross. Thus 1-planar graphs have geometric thickness two. In addition, the drawing is nearly 1-planar, that is, it is 1-planar if all fan-crossed edges are removed. An edge is fan-crossed if it is crossed by edges with a common vertex if it is crossed more than twice. The drawing algorithm uses high precision arithmetic with numbers with $O(n\log n)$ digits and computes the straight-line drawing from a 1-planar drawing in linear time on a real RAM.

A 3-prismatoid is the convex hull of two convex polygons A and B which lie in parallel planes $H_A,H_B\subset R^3$. Let $\tilde{A}$ be the orthogonal projection of A onto $H_B$. A 3-prismatoid is called nested if $\tilde{A}$ is properly contained in B, or vice versa. We show that every nested 3-prismatoid has an edge-unfolding to a non-overlapping polygon in the plane.

The Gromov-Hausdorff distance $\left(d_{GH}\right)$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for $X,Y\subset R^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension $d\ge 2$, $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$. For $d=1$, however, we prove that $d_{H,iso}\le \frac{5}{4}{d}_{GH}$. We also show that the bound is tight. In effect, for $X,Y\subset R$ with at most n points, this gives rise to an $O(n\log n)$-time algorithm to approximate $d_{GH}(X,Y)$ with an approximation factor of $(1+\frac{1}{4})$.

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees. As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on $n+1$ leaf nodes fall into $(2n-1)!!$ distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., $(n+1)!n!2^{-n}$. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.