We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs with m edges and n vertices only rather simple O(mn)-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time O(n 2 ). For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an O(m 2 / log n)-time algorithm for 2-edge strongly connected components, and thus improve over the O(mn) running time also when m = O(n). Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of O(n 2 log n) for k-edge-connectivity and O(n 3 ) for k-vertex-connectivity.improve upon the previous O(mn)-time algorithms for both 2eSCCs [NW93, GIL + 15a] and 2vSCCs [Jab14]. For 2eSCCs the previous upper bound stood for 20 years. Our approach immediately generalizes to computing the k-edge strongly connected components (keSCCs) and the k-vertex strongly connected components (kvSCCs). We give algorithms that, for any integral constant k > 2, compute (1) the keSCCs in time O(n 2 log n) (improving upon the previous upper bound of O(mn) [NW93]) and (2) the kvSCCs in time O(n 3 ) (improving upon the previous upper bound of O(mn 2 ) [Mak88]).Related Work. The 2-edge and 2-vertex connected components of an undirected graph can be determined in linear time [Tar72,HT73]. In directed graphs several related problems can be solved in linear time: Testing whether a graph is 2-edge or 2-vertex strongly connected [Tar76, GT85, Geo10], finding all strong bridges and strong articulation points [ILS12], and determining the 2-edge and 2-vertex strongly connected blocks [GIL + 15a, GIL + 15b]. An edge is a strong bridge and a vertex is a strong articulation point, respectively, if its removal from the graph increases the number of strongly connected components (SCCs) of the graph. Note the difference between 'blocks' and 'components' in directed graphs: In a 2-edge strongly connected block every pair of distinct vertices is 2-edge strongly connected; however, as opposed to a 2eSCC, the paths to connect the vertices in a block might use vertices that are not in the same block. Each 2eSCC is completely contained in one 2-edge strongly connected block, i.e., the 2eSCCs refine the 2-edge strongly connected blocks. In Appendix C we provide a construction that shows that knowing the 2-edge strongly connected blocks of a graph does not help in finding its 2eSCCs. The relation between blocks and components for vertex connectivity is analogous.Georgiadis et al. [GIL + 15a] and Jaberi [Jab14] described simple algorithms to compute the 2eSCCs and 2vSCCs in O(mn)-time, respectively, and posed as an open problem whether this can be improved to linear time as well. An O(mn) running time for computing the 2eSCCs was already achieved by Nagamochi and Watanabe...

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(n 2 ) time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time O(m 3/2 ), which further improves the running times for * Work partially done while visiting the Università di Roma "Tor Vergata".

A model of computation that is widely used in the formal analysis of reactive systems is symbolic algorithms. In this model the access to the input graph is restricted to consist of symbolic operations, which are expensive in comparison to the standard RAM operations. We give lower bounds on the number of symbolic operations for basic graph problems such as the computation of the strongly connected components and of the approximate diameter as well as for fundamental problems in model checking such as safety, liveness, and coliveness. Our lower bounds are linear in the number of vertices of the graph, even for constant-diameter graphs. For none of these problems lower bounds on the number of symbolic operations were known before. The lower bounds show an interesting separation of these problems from the reachability problem, which can be solved with O(D) symbolic operations, where D is the diameter of the graph.Additionally we present an approximation algorithm for the graph diameter which requiresÕ(n √ D) symbolic steps to achieve a (1 + )-approximation for any constant > 0. This compares to O(n · D) symbolic steps for the (naive) exact algorithm and O(D) symbolic steps for a 2-approximation. Finally we also give a refined analysis of the strongly connected components algorithms of [15], showing that it uses an optimal number of symbolic steps that is proportional to the sum of the diameters of the strongly connected components.

Abstract. The computation of the winning set for parity objectives and for Streett objectives in graphs as well as in game graphs are central problems in computer-aided verification, with application to the verification of closed systems with strong fairness conditions, the verification of open systems, checking interface compatibility, well-formedness of specifications, and the synthesis of reactive systems. We show how to compute the winning set on n vertices for (1) parity-3 (aka one-pair Streett) objectives in game graphs in time O(n 5/2 ) and for (2) k-pair Streett objectives in graphs in time O(n 2 + nk log n). For both problems this gives faster algorithms for dense graphs and represents the first improvement in asymptotic running time in 15 years.