Related papers: Exact Subquadratic Algorithm for Many-to-Many Matc…
Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair.…
Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is…
A bottleneck plane perfect matching of a set of $n$ points in $\mathbb{R}^2$ is defined to be a perfect non-crossing matching that minimizes the length of the longest edge; the length of this longest edge is known as {\em bottleneck}. The…
Given two point sets S and T, we study the many-to-many matching with demands problem (MMD problem) which is a generalization of the many-to-many matching problem (MM problem). In an MMD, each point of one set must be matched to a given…
We present subquadratic algorithms in the algebraic decision-tree model for several \textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise…
We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set $S$ of $n$ points in the plane and the goal is to find two smallest congruent disks whose union contains all points…
Let A={a_1,a_2,...,a_s} and {b_1,b_2,...,b_t} with s+r=n, the many to many point matching with demands and capacities matches each point a_i in A to at least alpha_i and at most alpha_i points in B, and each point b_j in B to at least…
This paper studies the problem of enumerating all maximal collinear subsets of size at least three in a given set of $n$ points. An algorithm for this problem, besides solving degeneracy testing and the exact fitting problem, can also help…
Given a set $ P $ of $n$ points and a set $ H $ of $n$ half-planes in the plane, we consider the problem of computing a smallest subset of points such that each half-plane contains at least one point of the subset. The previously best…
A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an approximate…
We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct:…
Given two point sets S and T, in a many-to-many matching between S and T each point in S is assigned to one or more points in T and vice versa. A generalization of the many-to-many matching problem is the limited capacity many-to-many…
The (unweighted) point-separation problem asks, given a pair of points $s$ and $t$ in the plane, and a set of candidate geometric objects, for the minimum-size subset of objects whose union blocks all paths from $s$ to $t$. Recent work has…
We study the problem of estimating the size of maximum matching and minimum vertex cover in sublinear time. Denoting the number of vertices by $n$ and the average degree in the graph by $\bar{d}$, we obtain the following results for both…
In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise…
We consider a bichromatic two-center problem for pairs of points. Given a set $S$ of $n$ pairs of points in the plane, for every pair, we want to assign a red color to one point and a blue color to the other, in such a way that the value…
Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand…
Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time…
In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the…
We consider the planar Euclidean two-center problem in which given $n$ points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic $O(n \log n)$-time algorithm for the case…