Nikos Parotsidis

2-Edge Connectivity in Directed Graphs

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly, not much has been investigated for directed graphs. In this article, we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edge-disjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this article is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, when two query vertices v and w are not 2-edge-connected, we can produce in constant time a “witness” of this property by exhibiting an edge that is contained in all paths from v to w or in all paths from w to v. We are also able to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.

Centrality-Aware Link Recommendations

Link recommendations are critical for both improving the utility and expediting the growth of social networks. Most previous approaches focus on suggesting links that are highly likely to be adopted. In this paper, we add a different perspective to the problem by aiming at recommending links that also improve specific properties of the network. In particular, our goal is to recommend to users links that if adopted would improve their centrality in the network. Specifically, we introduce the centrality-aware link recommendation problem as the problem of recommending to a user u, k links from a pool of recommended links so as to maximize the expected decrease of the sum of the shortest path distances of

2-edge connectivity in directed graphs

u

2-connectivity in directed graphs: an experimental study

to all other nodes in the network. We show that the problem is NP-hard, but our optimization function is monotone and sub-modular which guarantees a constant approximation ratio for the greedy algorithm. We present a fast algorithm for computing the expected decrease caused by a set of recommendations which we use as a building block in our algorithms. We provide experimental results that evaluate the performance of our algorithms with respect to both the accuracy of the prediction and the improvement in the centrality of the nodes, and we study the tradeoff between the two.

Loop Nesting Forests, Dominators, and Applications

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edge-disjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this paper is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, we also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.

Dominator Certification and Independent Spanning Trees: An Experimental Study

Graph connectivity is a fundamental concept in graph theory with numerous practical applications. Very recently, various notions of 2-connectivity in directed graphs (digraphs) have been introduced. In particular, 2-connectivity revealed to have a much richer and more complicated structure in directed graphs than in undirected graphs. In this paper we consider the computation of the 2-connected components and the 2-connected blocks of a digraph in practice, in the case of both edge and vertex connectivity. Specifically, we present efficient implementations of previously proposed and of new algorithms for computing the 2-vertex-connected components and the 2-vertex-connected blocks, the 2-edge-connected components and the 2-edge-connected blocks, and evaluate their performance experimentally on large digraphs taken from a variety of application areas. To the best of our knowledge, this is the-first empirical study for these problems. Our extensive experimental study sheds light on the relative difficulty of computing these notions of 2-connectivity in digraphs in practice. Furthermore, our experimental results suggest that the 2-vertex- and 2-edge-connected components of digraphs that arise in many practical applications can be found efficiently, despite the fact that currently the best known asymptotical bound for their computation is O(mn).

Strong connectivity in directed graphs under failures, with applications

Loop nesting forests and dominator trees are important tools in program optimization and code generation, and they have applications in other diverse areas. In this work we first present carefully engineered implementations of efficient algorithms for computing a loop nesting forest of a given directed graph, including a very efficient algorithm that computes the forest in a single depth-first search. Then we revisit the problem of computing dominators and present efficient implementations of the algorithms recently proposed by Fraczak et al. [12], which include an algorithm for acyclic graphs and an algorithm that computes both the dominator tree and a loop nesting forest. We also propose a new algorithm than combines the algorithm of Fraczak et al. for acyclic graphs with the algorithm of Lengauer and Tarjan. Finally, we provide fast algorithms for the following related problems: computing bridges and testing 2-edge connectivity, verifying dominators and testing 2-vertex connectivity, and computing a low-high order and two independent spanning trees. We exhibit the efficiency of our algorithms experimentally on large graphs taken from a variety of application areas.

Faster algorithms for computing maximal 2-connected subgraphs in sparse directed graphs

We present the first implementations of certified algorithms for computing dominators, and exhibit their efficiency experimentally on graphs taken from a variety of applications areas. The certified algorithms are obtained by augmenting dominator-finding algorithms to compute a certificate of correctness that is easy to verify. A suitable certificate for dominators is obtained from the concepts of low-high orders and independent spanning trees. Therefore, our implementations provide efficient constructions of these concepts as well, which are interesting in their own right. Furthermore, we present an experimental study of efficient algorithms for computing dominators on large graphs.

2-Vertex Connectivity in Directed Graphs

Let G be a directed graph (digraph) with m edges and n vertices, and let G \ e (resp., G \ v) be the digraph obtained after deleting edge e (resp., vertex v) from G. We show how to compute in O(m + n) worst-case time: • The total number of strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. • The size of the largest and of the smallest strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. Let G be strongly connected. We say that edge e (resp., vertex v) separates two vertices x and y, if x and y are no longer strongly connected in G \ e (resp., G \ v). We also show how to build in O(m+n) time O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: • Report in O(n) worst-case time all the strongly connected components of G \ e (resp., G \ v), for a query edge e (resp., vertex v). • Test whether an edge or a vertex separates two query vertices in O(1) worst-case time. • Report all edges (resp., vertices) that separate two query vertices in optimal worst-case time, i.e., in time O(k), where k is the number of separating edges (resp., separating vertices). (For k = 0, the time is O(1)). All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by 1-edge and 1-vertex cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts.

Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with O(mn) time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced O(n2) time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time O(m3/2), which further improves the running times for sparse graphs. The notion of 2-connectivity naturally generalizes to k-connectivity for k > 2. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time O(m3/2 log n), improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in O(n2 log n) time.