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In this work, we study the connected version of the covering problem motivated by the coverage of ad-hoc drones’ swarm. We focus on the situation where the number of drones is given, and this number is not necessarily enough to cover all users. That is, we deal with a budget optimization problem, where the budget is the number of given drones. We assume that each ground user has different QoS requirements. Additionally, each ground user has a weight that corresponds to the importance (rank) of the user. Moreover, we consider the case when there is no third-party entity that provides connectivity to the drones. In this paper, we propose a 3D deployment scheme with the given number of drones such that the sum of the weights (ranks) of the ground users covered by drones is maximized (when the covering radii satisfy QoS of these users), and the drones form a connected graph. We present a number of approximate solutions with provable guaranteed performance evaluation that have been validated also through the simulation platform.

Abstract: A graph is called {\em $\el$-connected from $U$ to $r$} if there are $\el$ internally disjoint paths from every node $u \in U$ to $r$. The {\em Rooted Subset Connectivity Augmentation Problem} ({\em RSCAP}) is as follows: given a graph $G=(V+r,E)$, a node subset $U \subseteq V$, and an integer $k$, find a smallest set $F$ of new edges such that $G+F$ is $k$-connected from $U$ to $r$. In this paper we consider mainly a restricted version of RSCAP in which the input graph $G$ is already $(k-1)$-connected from $U$ to $r$. For this version we give an $O(\ln\! |U|)$-approximation algorithm, and show that the problem cannot achieve a better approximation guarantee than the Set Cover Problem (SCP) on $|U|$ elements and with $|V|-|U|$ sets. For the general version of RSCAP we give an $O(\ln k \ln\!|U|)$-approximation algorithm. For $U=V$ we get the {\em Rooted Connectivity Augmentation Problem} ({\em RCAP}). For directed graphs RCAP is polynomially solvable, but for undirected graphs its complexity status is not known: no polynomial algorithm is known, and it is also not known to be NP-hard. For undirected graphs with the input graph $G$ being $(k-1)$-connected from $V$ to $r$, we give an algorithm that computes a solution of size at most ${\it opt}+\min\{opt,k\}/2$, where {\it opt} denotes the optimal solution size.

Given a graph (directed or undirected) with costs on the edges, and an integer $k$, we consider the problem of finding a $k$-node connected spanning subgraph of minimum cost. For the general instance of the problem (directed or undirected), there is a simple $2k$-approximation algorithm. Better algorithms are known for various ranges of $n,k$. For undirected graphs with metric costs Khuller and Raghavachari gave a $( 2+{2(k-1)}/{n})$-approximation algorithm. We obtain the following results: (i) For arbitrary costs, a $k$-approximation algorithm for undirected graphs and a $(k+1)$-approximation algorithm for directed graphs. (ii) For metric costs, a $(2+({k-1})/{n})$-approximation algorithm for undirected graphs and a $(2+{k}/{n})$-approximation algorithm for directed graphs. For undirected graphs and $k=6,7$, we further improve the approximation ratio from $k$ to $\lceil (k+1)/2 \rceil=4$; previously, $\lceil (k+1)/2 \rceil$-approximation algorithms were known only for $k \leq 5$. We also give a fast $3$-approximation algorithm for $k=4$. The multiroot problem generalizes the min-cost $k$-connected subgraph problem. In the multiroot problem, requirements $k_u$ for every node $u$ are given, and the aim is to find a minimum-cost subgraph that contains $\max\{k_u,k_v\}$ internally disjoint paths between every pair of nodes $u,v$. For the general instance of the problem, the best known algorithm has approximation ratio $2k$, where $k=\max k_u$. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve for $k \leq 7$ the approximation guarantee from $3$ to $2+{\lfloor (k-1)/2 \rfloor}/{k} < 2.5$.

Consider the following problem: Given an undirected graph with nonnegative edge costs and requirements ku for every node u, find a minimum-cost subgraph that contains max{ku, kv} internally disjoint paths between every pair of nodes u, v. For k = max ku ≥ 2, this problem is NP-hard. The best-known algorithm for it has an approximation ratio of 2(k − 1). For a general instance of the problem, for no value of k ≥ 2, a better approximation algorithm was known. We consider the case of small requirements ku ∈ {1, 2, 3}; these may arise in applications, as, in practical networks, the connectivity requirements are usually rather small. For this case, we give an algorithm with an approximation ratio of ${\bf 10}\over{\bf 3}$. This improves the best previously known approximation ratio of 4. Our algorithm also implies an improvement for arbitrary k. In the case in which we have an initial graph which is 2-connected, our algorithm achieves an approximation ratio of 2. © 2000 John Wiley & Sons, Inc.

Given a digraph D, the minimum integral dicycle cover problem (known also as the minimum feedback arc set problem) is to find a minimum set of arcs that intersects every dicycle; the maximum integral dicycle packing problem is to find a maximum set of pairwise arc disjoint dicycles. These two problems are NP-complete. Assume D has a 2-vertex cut. We show how to derive a minimum dicycle cover (a maximum dicycle packing) for D, by composing certain covers (packings) of the corresponding pieces. The composition of the covers is simple and was partially considered in the literature before. The main contribution of this paper is to the packing problem. Let τ be the value of a minimum integral dicycle cover, and ν* (ν) the value of a maximum (integral) dicycle packing. We show that if τ = ν then a simple composition, similar to that of the covers, is valid for the packing problem. We use these compositions to extend an O(n3) (resp., O(n4)) algorithm for finding a minimum integral dicycle cover (resp., packing) from planar digraphs to K3,3-free digraphs (i.e., digraphs not containing any subdivision of K3,3). However, if τ ≠ ν, then such a simple composition for the packing problem is not valid. We show, that if the pieces satisfy, what we call, the stability property, then a simple composition does work. We prove that if ν = ν* holds for each piece, then the stability property holds as well. Further, we use the stability property to show that if ν = ν* holds for each piece, then ν = ν* holds for D as well.

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G=(V,E) is considered. For k≥2, this problem is known to be NP-hard. Combining properties of inclusion-minimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist kinternally vertex-disjoint paths to every other vertex), we derive polynomial time algorithm for finding a (⌈k/2⌉+1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k=3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V|3|E|)=O(|V|5).

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G=(V,E) is considered. For k≥2, this problem is known to be NP-hard. Based on the paper of Auletta, Dinitz, Nutov, and Parente in this issue, we derive a 3-approximation algorithm for k∈{4,5}. This improves the best previously known approximation ratios 416and 41730, respectively. The complexity of the suggested algorithm is O(|V|5) for the deterministic and O(|V|4log|V|) for the randomized version. The way of solution is as follows. Analyzing a subgraph constructed by the algorithm of the aforementioned paper, we prove that all its “small” cuts have exactly two sides and separate a certain fixed pair of vertices. Such a subgraph is augmented to a k-connected one (optimally) by at most four executions of a min-cost k-flow algorithm.