We consider network design problems in which we are given a graph and seek a min-size $2$-connected subgraph that satisfies a prescribed property.
$\bullet$ In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a min-size edge subset of a specified edge set such that the augmented graph is $2$-connected. We breach the natural ratio of $2$ for this problem and also for the more general Crossing Family Cover problem.
$\bullet$ In the $2$-Connected Dominating Set problem we seek a minimum size $2$-connected subgraph that dominates all nodes. We give the first non-trivial approximation algorithm for this problem, with expected ratio $O(σ\log^3 n)$, where $σ=O(\log n \cdot\log\log n\cdot(\log\log\log n)^{3})$.
The unifying technique of both results is a reduction to the Subset Steiner Connected Dominating Set problem. Such a reduction was known for edge-connectivity, and we extend it to $2$-node connectivity problems. We show that the same method can be used to obtain easily polylogarithmic approximation ratios that are not too far from the best known ones for several other problems.
We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and $k$-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node $v$ in the solution is at most $2b(v) + O(k)$ where $b(v)$ is the degree upper bound on $v$. For undirected graphs and element-connectivity requirements with maximum connectivity…
The terminal backup problems (Anshelevich and Karagiozova (2011)) form a class of network design problems: Given an undirected graph with a requirement on terminals, the goal is to find a minimum cost subgraph satisfying the connectivity requirement. The node-connectivity terminal backup problem requires a terminal to connect other terminals with a number of node-disjoint paths. This problem is not known whether is NP-hard or tractable. Fukunaga (2016) gave a $4/3$-approximation algorithm based on LP-rounding scheme using a general LP-solver. In this paper, we develop a combinatorial algorithm…
A graph $G$ is $k$-out-connected from its node $s$ if it contains $k$ internally disjoint $sv$-paths to every node $v$; $G$ is $k$-connected if it is $k$-out-connected from every node. In connectivity augmentation problems the goal is to augment a graph $G_0=(V,E_0)$ by a minimum costs edge set $J$ such that $G_0 \cup J$ has higher connectivity than $G_0$. In the $k$-Out-Connectivity Augmentation ($k$-OCA) problem, $G_0$ is $(k-1)$-out-connected from $s$ and $G_0 \cup J$ should be $k$-out-connected from $s$; in the $k$-Connectivity Augmentation ($k$-CA) problem $G_0$ is $(k-1)$-connected and $…
Many network design problems deal with the design of low-cost networks that are resilient to the failure of their elements, such as nodes or links. One such problem is Connectivity Augmentation, where the goal is to cheaply increase the connectivity of a network from a value k to k+1. The most studied setting focuses on edge-connectivity, which reduces to k=2, called Cactus Augmentation. From an approximation perspective, Byrka, Grandoni, and Jabal Ameli (2020) were the first to break the 2-approximation barrier for this problem, by exploiting a connection to the Steiner Tree problem, and by t…
We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph $G=(V,E)$ and integer connectivity requirements $r(uv)$ for each unordered pair of nodes $uv$. The goal is to find a minimum weighted subgraph $H$ of $G$ such that $H$ contains $r(uv)$ disjoint paths between $u$ and $v$ for each node pair $uv$. Three versions of the problem are edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP) and vertex-connectivity SNDP (VC-SNDP) depending on whether the paths are require…
In this article we discuss general strategies and computer algorithms to test the connectivity of unstructured networks which consist of a number of segments connected through randomly distributed nodes.
In this paper, we study a generalization of the classical minimum cut prob- lem, called Connectivity Preserving Minimum Cut (CPMC) problem, which seeks a minimum cut to separate a pair (or pairs) of source and destination nodes and meanwhile ensure the connectivity between the source and its partner node(s). The CPMC problem is a rather powerful formulation for a set of problems and finds applications in many other areas, such as network security, image processing, data mining, pattern recognition, and machine learning. For this important problem, we consider two variants, connectiv- ity prese…
A subset $T \subseteq V$ of terminals is $k$-connected to a root $s$ in a directed/undirected graph $J$ if $J$ has $k$ internally-disjoint $vs$-paths for every $v \in T$; $T$ is $k$-connected in $J$ if $T$ is $k$-connected to every $s \in T$. We consider the {\sf Subset $k$-Connectivity Augmentation} problem: given a graph $G=(V,E)$ with edge/node-costs, node subset $T \subseteq V$, and a subgraph $J=(V,E_J)$ of $G$ such that $T$ is $k$-connected in $J$, find a minimum-cost augmenting edge-set $F \subseteq E \setminus E_J$ such that $T$ is $(k+1)$-connected in $J \cup F$. The problem admits tr…
This paper examines the problem of assigning a transmission power to every node of a wireless sensor network. The goal is to minimize the total power consumption while ensuring that the resulting communication graph is connected. We focus on a restricted version of this Range Assignment (RA) problem in which there are two different power levels. We only consider symmetrical transmission links to allow easy integration with low level wireless protocols that typically require bidirectional communication between two neighboring nodes. We introduce a parameterized polynomial time approximation alg…
We present deterministic constant-round protocols for the graph connectivity problem in the model where each of the $n$ nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifically, we exhibit a deterministic protocol that computes the connected components of the input graph in $\lceil 1/ε\rceil$ rounds, each player communicating $\mathcal{O}(n^ε \cdot \log n)$ bits per round, with $0 < ε\leq 1$. We also provide a deter…
We consider the problem of reliably broadcasting information in a multihop asyn- chronous network that is subject to Byzantine failures. That is, some nodes of the network can exhibit arbitrary (and potentially malicious) behavior. Existing solutions provide de- terministic guarantees for broadcasting between all correct nodes, but require that the communication network is highly-connected (typically, 2k + 1 connectivity is required, where k is the total number of Byzantine nodes in the network). In this paper, we investigate the possibility of Byzantine tolerant reliable broadcast be- tween m…
Given a network in which some pairs of nodes can communicate freely, and some subsets of the nodes could be faulty and colluding to disrupt communication, when can messages reliably be sent from one given node to another? We give a new characterization of when the agreement problem can be solved and provide an agreement algorithm which can reach agreement when the number of Byzantine nodes along each minimal vertex cut is bounded. Our new bound holds for a strict superset of cases than the previously known bound. We show that the new bound is tight. Furthermore, we show that this algorithm doe…
The congested clique is a synchronous, message-passing model of distributed computing in which each computational unit (node) in each round can send message of O(log n) bits to each other node of the network, where n is the number of nodes. This model has been considered under two extreme scanarios: unicast or broadcast. In the unicast model, a node can send (possibly) different message to each other node of the network. In contrast, in the broadcast model each node sends a single (the same) message to all other nodes. We study the congested clique model parametrized by the range r, the maximu…
Self-stabilization for non-masking fault-tolerant distributed system has received considerable research interest over the last decade. In this paper, we propose a self-stabilizing algorithm for 2-edge-connectivity and 2-vertex-connectivity of an asynchronous distributed computer network. It is based on a self-stabilizing depth-first search, and is not a composite algorithm in the sense that it is not composed of a number of self-stabilizing algorithms that run concurrently. The time and space complexities of the algorithm are the same as those of the underlying self-stabilizing depth-first sea…
We consider the problem of constructing a communication infrastructure from scratch, for a collection of identical wireless nodes. Combinatorially, this means a) finding a set of links that form a strongly connected spanning graph on a set of $n$ points in the plane, and b) scheduling it efficiently in the SINR model of interference. The nodes must converge on a solution in a distributed manner, having no means of communication beyond the sole wireless channel. We give distributed connectivity algorithms that run in time $O(poly(\log Δ, \log n))$, where $Δ$ is the ratio between the longest and…
We consider a Min-Power Bounded-Hops Symmetric Connectivity problem that consists in the construction of communication spanning tree on a given graph, where the total energy consumption spent for the data transmission is minimized and the maximum number of hops between two nodes is bounded by some predefined constant. We focus on the planar Euclidian case of this problem where the nodes are placed at the random uniformly spread points on a square and the power cost necessary for the communication between two network elements is proportional to the squared distance between them. Since this is a…
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
In this paper, we present algorithms for designing networks that are robust to node failures with minimal or limited number of links. We present algorithms for both the static network setting and the dynamic network setting; setting where new nodes can arrive in the future. For the static setting, we present algorithms for constructing the optimal network in terms of the number of links used for a given node size and the number of nodes that can fail. We then consider the dynamic setting where it is disruptive to remove any of the older links. For this setting, we present online algorithms for…
