Incremental 22-Edge-Connectivity in Directed Graphs

In this paper, we initiate the study of the dynamic maintenance of $2$-edge-connectivity relationships in directed graphs. We present an algorithm that can update the $2$-edge-connected blocks of a directed graph with $n$ vertices through a sequence of $m$ edge insertions in a total of $O(mn)$ time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices $v$ and $w$ are $2$-edge-connected. Moreover, if $v$ 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 $v$. (ii) Report in $O(n)$ time all the $2$-edge-connected blocks of $G$. To the best of our knowledge, this is the first dynamic algorithm for $2$-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.Comment: Full version of paper presented at ICALP 201

Faster Algorithms for Computing Maximal 2-Connected Subgraphs in Sparse 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 $\widetilde{O}(mn)$ 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 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 $O(m^{3/2} \log{n})$, improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in $O(n^2 \log n)$ time.Comment: Revised version of SODA 2017 paper including details for k-edge-connected subgraph

Approximating subset kk-connectivity problems

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 trivial ratio $O(|T|^2)$. We consider the case $|T|>k$ and prove that for directed/undirected graphs and edge/node-costs, a $\rho$-approximation for {\sf Rooted Subset $k$-Connectivity Augmentation} implies the following ratios for {\sf Subset $k$-Connectivity Augmentation}: (i) $b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k})$; (ii) $\rho \cdot O(\frac{|T|}{|T|-k} \log k)$, where b=1 for undirected graphs and b=2 for directed graphs, and $H(k)$ is the $k$th harmonic number. The best known values of $\rho$ on undirected graphs are $\min\{|T|,O(k)\}$ for edge-costs and $\min\{|T|,O(k \log |T|)\}$ for node-costs; for directed graphs $\rho=|T|$ for both versions. Our results imply that unless $k=|T|-o(|T|)$, {\sf Subset $k$-Connectivity Augmentation} admits the same ratios as the best known ones for the rooted version. This improves the ratios in \cite{N-focs,L}

Hamilton cycles in bidirected complete graphs

Zaslavsky observed that the topics of directed cycles in directed graphs and alternating cycles in edge 2-colored graphs have a common generalization in the study of coherent cycles in bidirected graphs. There are classical theorems by Camion, Harary and Moser, Häggkvist and Manoussakis, and Saad which relate strong connectivity and Hamiltonicity in directed "complete" graphs and edge 2-colored "complete" graphs. We prove two analogues to these theorems for bidirected "complete" signed graphs

Faster Algorithms for Rooted Connectivity in Directed Graphs

We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time $\tilde{O}(n^2)$. For rooted edge connectivity this is the first algorithm to improve on the $\Omega(n^3)$ time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems. We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a $(1 + \epsilon)$-approximation for rooted vertex connectivity in $\tilde{O}(nW/\epsilon)$ time where $W$ is the total vertex weight (assuming integral vertex weights); in particular this yields an $\tilde{O}(n^2/\epsilon)$ time randomized algorithm for unweighted graphs. This translates to a $\tilde{O}(\kappa nW)$ time exact algorithm where $\kappa$ is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity. Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [9] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [24] and Forster et al. [7] for vertex connectivity

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

Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known results. For \textsf{2EC-B} and \textsf{2EC-B-C}, we present new $4$-approximation algorithms that run in linear time. We also propose various heuristics to improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios

Computing vertex-edge cut-pairs and 2-edge cuts in practice

4sìopenWe consider two problems regarding the computation of connectivity cuts in undirected graphs, namely identifying vertex-edge cut-pairs and identifying 2-edge cuts, and present an experimental study of efficient algorithms for their computation. In the first problem, we are given a biconnected graph G and our goal is to find all vertices v such that G v is not 2-edge-connected, while in the second problem, we are given a 2-edge-connected graph G and our goal is to find all edges e such that G e is not 2-edge-connected. These problems are motivated by the notion of twinless strong connectivity in directed graphs but are also of independent interest. Moreover, the computation of 2-edge cuts is a main step in algorithms that compute the 3-edge-connected components of a graph. In this paper, we present streamlined versions of two recent linear-time algorithms of Georgiadis and Kosinas that compute all vertex-edge cut-pairs and all 2-edge cuts, respectively. We compare the empirical performance of our vertex-edge cut-pairs algorithm with an alternative linear-time method that exploits the structure of the triconnected components of G. Also, we compare the empirical performance of our 2-edge cuts algorithm with the algorithm of Tsin, which was reported to be the fastest one among the previously existing for this problem. To that end, we conduct a thorough experimental study to highlight the merits and weaknesses of each technique.openGeorgiadis L.; Giannis K.; Italiano G.F.; Kosinas E.Georgiadis, L.; Giannis, K.; Italiano, G. F.; Kosinas, E