ﻻ يوجد ملخص باللغة العربية
Finding cohesive subgraphs in a network is a well-known problem in graph theory. Several alternative formulations of cohesive subgraph have been proposed, a notable example being $s$-club, which is a subgraph where each vertex is at distance at most $s$ to the others. Here we consider the problem of covering a given graph with the minimum number of $s$-clubs. We study the computational and approximation complexity of this problem, when $s$ is equal to 2 or 3. First, we show that deciding if there exists a cover of a graph with three $2$-clubs is NP-complete, and that deciding if there exists a cover of a graph with two $3$-clubs is NP-complete. Then, we consider the approximation complexity of covering a graph with the minimum number of $2$-clubs and $3$-clubs. We show that, given a graph $G=(V,E)$ to be covered, covering $G$ with the minimum number of $2$-clubs is not approximable within factor $O(|V|^{1/2 -varepsilon})$, for any $varepsilon>0$, and covering $G$ with the minimum number of $3$-clubs is not approximable within factor $O(|V|^{1 -varepsilon})$, for any $varepsilon>0$. On the positive side, we give an approximation algorithm of factor $2|V|^{1/2}log^{3/2} |V|$ for covering a graph with the minimum number of $2$-clubs.
Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such flipping of the
A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The
The bin covering problem asks for covering a maximum number of bins with an online sequence of $n$ items of different sizes in the range $(0,1]$; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the
We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the pr
In this paper, we study the lower- and upper-bounded covering (LUC) problem, where we are given a set $P$ of $n$ points, a collection $mathcal{B}$ of balls, and parameters $L$ and $U$. The goal is to find a minimum-sized subset $mathcal{B}subseteq ma