相关论文: Integer Point Sets Minimizing Average Pairwise L1-…

In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [\Delta]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that…

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

The n-way number partitioning problem, a fundamental challenge in combinatorial optimization, has significant implications for applications such as fair division and machine scheduling. Despite these problems being NP-hard, many…

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and…

A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent…

According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log…

For three decades, carrier-phase observations have been used to obtain the most accurate location estimates using global navigation satellite systems (GNSS). These estimates are computed by minimizing a nonlinear mixed-integer least-squares…

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have.…

Is it possible to draw a circle in Manhattan, using only its discrete network of streets and boulevards? In this study, we will explore the construction and properties of circular paths on an integer lattice, a discrete space where the…

A balanced V-shape is a polygonal region in the plane contained in the union of two crossing equal-width strips. It is delimited by two pairs of parallel rays that emanate from two points x, y, are contained in the strip boundaries, and are…

The purpose of this paper is to introduce an algorithm that can detect the most unusual part of a digital image in probabilistic setting. The most unusual part of a given shape is defined as a part of the image that has the maximal distance…

The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which…

Expected urban population doubling calls for a compelling theory of the city. Random walks and diffusions defined on spatial city graphs spot hidden areas of geographical isolation in the urban landscape going downhill. First--passage time…

As cities struggle to adapt to more ``people-centered'' urbanism, transportation planning and engineering must innovate to expand the street network strategically in order to ensure efficiency but also to deter sprawl. Here, we conducted a…

We propose the first algorithm for non-rigid 2D-to-3D shape matching, where the input is a 2D shape represented as a planar curve and a 3D shape represented as a surface; the output is a continuous curve on the surface. We cast the problem…

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let $S_n$ denote the set of permutations on $n$ symbols, and for each $\sigma, \tau \in S_n$, define their Ulam distance…

Computer vision methods that quantify the perception of urban environment are increasingly being used to study the relationship between a city's physical appearance and the behavior and health of its residents. Yet, the throughput of…

The basic question that is addressed in this paper is finding the closest separable state for a given entangled state, measured with the Hilbert Schmidt distance. While this problem is in general very hard, we show that the following…

The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness…