arXiv subjects

A. Karim Abu-Affash

Publications and source records attributed to A. Karim Abu-Affash.

11 recordsLinked to original sources

arXiv2026

The Minimum Dominating Set Problem on Bipartite Circle Graphs: Complexity and Approximation

A circle graph is the intersection graph of a set of chords in a circle. A dominating set of a graph $G=(V,E)$ is a subset $D\subseteq V$ such that every vertex in $V\setminus D$ is adjacent to at least one vertex of $D$. Computing a minimum dominating set is known to be NP-hard on circle graphs. In this paper, we study the minimum dominating set problem on bipartite circle graphs, namely, circle graphs admitting a chord representation in which the chords can be partitioned into two color classes such that no two chords of the same color intersect. We prove that the problem remains NP-hard for this restricted graph class by a reduction from Planar Monotone 3-SAT. On the positive side, we present a polynomial-time 2-approximation algorithm and develop a polynomial-time approximation scheme (PTAS) based on local search.

cs.CG↗

arXiv2023

Geometric Spanning Trees Minimizing the Wiener Index

The Wiener index of a network, introduced by the chemist Harry Wiener, is the sum of distances between all pairs of nodes in the network. This index, originally used in chemical graph representations of the non-hydrogen atoms of a molecule, is considered to be a fundamental and useful network descriptor. We study the problem of constructing geometric networks on point sets in Euclidean space that minimize the Wiener index: given a set $P$ of $n$ points in $\mathbb{R}^d$, the goal is to construct a network, spanning $P$ and satisfying certain constraints, that minimizes the Wiener index among the allowable class of spanning networks. In this work, we focus mainly on spanning networks that are trees and we focus on problems in the plane ($d=2$). We show that any spanning tree that minimizes the Wiener index has non-crossing edges in the plane. Then, we use this fact to devise an $O(n^4)$-time algorithm that constructs a spanning tree of minimum Wiener index for points in convex position. We also prove that the problem of computing a spanning tree on $P$ whose Wiener index is at most $W$, while having total (Euclidean) weight at most $B$, is NP-hard. Computing a tree that minimizes the Wiener index has been studied in the area of communication networks, where it is known as the optimum communication spanning tree problem.

cs.CG↗

arXiv2023

Dynamic Euclidean Bottleneck Matching

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 bottleneck matching in sublinear update time. In this work, we answer this question in the affirmative for points on a real line and for points in the plane with a bounded geometric spread. For a set $P$ of $n$ points on a line, we show that there exists a dynamic algorithm that maintains a bottleneck matching of $P$ and supports insertion and deletion in $O(\log n)$ time. Moreover, we show that a modified version of this algorithm maintains a minimum-weight matching with $O(\log n)$ update (insertion and deletion) time. Next, for a set $P$ of $n$ points in the plane, we show that a ($6\sqrt{2}$)-factor approximate bottleneck matching of $P_k$, at each time step $k$, can be maintained in $O(\logΔ)$ amortized time per insertion and $O(\logΔ + |P_k|)$ amortized time per deletion, where $Δ$ is the geometric spread of $P$.

cs.CG↗

arXiv2022

Piercing Diametral Disks Induced by Edges of Maximum Spanning Tree

Let $P$ be a set of points in the plane and let $T$ be a maximum-weight spanning tree of $P$. For an edge $(p,q)$, let $D_{pq}$ be the diametral disk induced by $(p,q)$, i.e., the disk having the segment $\overline{pq}$ as its diameter. Let $\cal{D_T}$ be the set of the diametral disks induced by the edges of $T$. In this paper, we show that one point is sufficient to pierce all the disks in $\cal{D_T}$, thus, the set $\cal{D_T}$ is Helly. Actually, we show that the center of the smallest enclosing circle of $P$ is contained in all the disks of $\cal{D_T}$, and thus the piercing point can be computed in linear time.

cs.CG↗

arXiv2021

Piercing Pairwise Intersecting Geodesic Disks by Five Points

Given a simple polygon $P$ on $n$ vertices, and a set $D$ of $m$ pairwise intersecting geodesic disks in $P$, we show that five points in $P$ are always sufficient to pierce all the disks in $D$. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer.

cs.CG↗

arXiv2020

A Linear-Time Algorithm for Minimum $k$-Hop Dominating Set of a Cactus Graph

Given a graph $G=(V,E)$ and an integer $k \ge 1$, a $k$-hop dominating set $D$ of $G$ is a subset of $V$, such that, for every vertex $v \in V$, there exists a node $u \in D$ whose hop-distance from $v$ is at most $k$. A $k$-hop dominating set of minimum cardinality is called a minimum $k$-hop dominating set. In this paper, we present linear-time algorithms that find a minimum $k$-hop dominating set in unicyclic and cactus graphs. To achieve this, we show that the $k$-dominating set problem on unicycle graph reduces to the piercing circular arcs problem, and show a linear-time algorithm for piercing sorted circular arcs, which improves the best known $O(n\log n)$-time algorithm.

cs.DS↗

arXiv2020

Minimizing Total Interference in Asymmetric Sensor Networks

The problem of computing a connected network with minimum interference is a fundamental problem in wireless sensor networks. Several models of interference have been studied in the literature. The most common model is the receiver-centric, in which the interference of a node $p$ is defined as the number of other nodes whose transmission range covers $p$. In this paper, we study the problem of assigning a transmission range to each sensor, such that the resulting network is strongly connected and the total interference of the network is minimized. For the one-dimensional case, we show how to solve the problem optimally in $O(n^3)$ time. For the two-dimensional case, we show that the problem is NP-complete and give a polynomial-time 2-approximation algorithm for the problem.

cs.CG↗

arXiv2020

Planar Bichromatic Bottleneck Spanning Trees

Given a set $P$ of $n$ red and blue points in the plane, a \emph{planar bichromatic spanning tree} of $P$ is a spanning tree of $P$, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Moreover, we present a polynomial-time $(8\sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $λ$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8\sqrt{2}λ$.

cs.CG↗

arXiv2015

Approximating the Bottleneck Plane Perfect Matching of a Point Set

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 problem of computing a bottleneck plane perfect matching has been proved to be NP-hard. We present an algorithm that computes a bottleneck plane matching of size at least $\frac{n}{5}$ in $O(n \log^2 n)$-time. Then we extend our idea toward an $O(n\log n)$-time approximation algorithm which computes a plane matching of size at least $\frac{2n}{5}$ whose edges have length at most $\sqrt{2}+\sqrt{3}$ times the bottleneck.

cs.CG↗

arXiv2012

Bottleneck Non-Crossing Matching in the Plane

Let $P$ be a set of $2n$ points in the plane, and let $M_{\rm C}$ (resp., $M_{\rm NC}$) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of $P$. We study the problem of computing $M_{\rm NC}$. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an $O(n^{1.5}\log^{0.5} n)$-time algorithm that computes a non-crossing matching $M$ of $P$, such that $bn(M) \le 2\sqrt{10} \cdot bn(M_{\rm NC})$, where $bn(M)$ is the length of a longest edge in $M$. An interesting implication of our construction is that $bn(M_{\rm NC})/bn(M_{\rm C}) \le 2\sqrt{10}$.

cs.CG↗

arXiv2010

An Approximation Algorithm for the Euclidean Bottleneck Steiner Tree Problem

Given two sets of points in the plane, $P$ of $n$ terminals and $S$ of $m$ Steiner points, a Steiner tree of $P$ is a tree spanning all points of $P$ and some (or none or all) points of $S$. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, $P$ and $S$, and a positive integer $k \le m$, find a bottleneck Steiner tree of $P$ with at most $k$ Steiner points. The problem has application in the design of wireless communication networks. We first show that the problem is NP-hard and cannot be approximated within factor $\sqrt{2}$, unless $P=NP$. Then, we present a polynomial-time approximation algorithm with performance ratio 2.

cs.CG↗