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Evangelos Kosinas

Publications and source records attributed to Evangelos Kosinas.

13 recordsLinked to original sources
arXiv2026

Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time

Computing edge-connected components in directed and undirected graphs is a fundamental and well-studied problem in graph algorithms. In a very recent breakthrough, Korhonen [STOC 2025] showed that for any fixed $k$, the $k$-edge connected components of an undirected graph can be computed in linear time. In contrast, the directed case remains significantly more challenging: linear-time algorithms are only known for $k \le 3$, and for any fixed $k > 3$, the best known bound for sparse or moderately dense graphs is still the $O(mn)$-time algorithm of Nagamochi and Watanabe (1993). In this paper, we break the $O(mn)$ barrier for all $k = o(n^{1/4}/\sqrt{\log{n}})$. We present a randomized algorithm that computes the $(k+2)$-edge-connected components of a $k$-edge-connected directed graph in $O(k^2 m \sqrt{n} \log n)$ time, for any~$k$. This constitutes the first improvement over the classic Nagamochi--Watanabe bound for any constant $k > 3$. Our approach introduces new structural insights into directed edge-cuts and combines these with both new and existing techniques. A central contribution of our work is a substantial simplification and generalization of the framework introduced in~\cite{GKPP:3ECC}, which achieved an $\widetilde{O}(m\sqrt{m})$ bound for computing the $3$-edge-connected components of a digraph. In addition, we develop a variant of our algorithm that achieves the same $O(m \sqrt{n} \log n)$ running time for computing the $4$-edge-connected components of a \emph{general} directed graph.

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arXiv2025

Efficient Contractions of Dynamic Graphs -- with Applications

A non-trivial minimum cut (NMC) sparsifier is a multigraph $\hat{G}$ that preserves all non-trivial minimum cuts of a given undirected graph $G$. We introduce a flexible data structure for fully dynamic graphs that can efficiently provide an NMC sparsifier upon request at any point during the sequence of updates. We employ simple dynamic forest data structures to achieve a fast from-scratch construction of the sparsifier at query time. Based on the strength of the adversary and desired type of time bounds, the data structure comes with different guarantees. Specifically, let $G$ be a fully dynamic simple graph with $n$ vertices and minimum degree $δ$. Then our data structure supports an insertion/deletion of an edge to/from $G$ in $n^{o(1)}$ worst-case time. Furthermore, upon request, it can return w.h.p. an NMC sparsifier of $G$ that has $O(n/δ)$ vertices and $O(n)$ edges, in $\hat{O}(n)$ time. The probabilistic guarantees hold against an adaptive adversary. Alternatively, the update and query times can be improved to $\tilde{O}(1)$ and $\tilde{O}(n)$ respectively, if amortized-time guarantees are sufficient, or if the adversary is oblivious. We discuss two applications of our data structure. First, it can be used to efficiently report a cactus representation of all minimum cuts of a fully dynamic simple graph. Using the NMC sparsifier we can w.h.p. build this cactus in worst-case time $\hat{O}(n)$ against an adaptive adversary. Second, our data structure allows us to efficiently compute the maximal $k$-edge-connected subgraphs of undirected simple graphs, by repeatedly applying a minimum cut algorithm on the NMC sparsifier. Specifically, we can compute w.h.p. the maximal $k$-edge-connected subgraphs of a simple graph with $n$ vertices and $m$ edges in $\tilde{O}(m+n^2/k)$ time which is an improvement for $k = Ω(n^{1/8})$ and works for fully dynamic graphs.

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arXiv2025

An Optimal $3$-Fault-Tolerant Connectivity Oracle

We present an optimal oracle for answering connectivity queries in undirected graphs in the presence of at most three vertex failures. Specifically, we show that we can process a graph $G$ in $O(n+m)$ time, in order to build a data structure that occupies $O(n)$ space, which can be used in order to answer queries of the form "given a set $F$ of at most three vertices, and two vertices $x$ and $y$ not in $F$, are $x$ and $y$ connected in $G\setminus F$?" in constant time, where $n$ and $m$ denote the number of vertices and edges, respectively, of $G$. The idea is to rely on the DFS-based framework introduced by Kosinas [ESA'23], for handling connectivity queries in the presence of multiple vertex failures. Our technical contribution is to show how to appropriately extend the toolkit of the DFS-based parameters, in order to optimally handle up to three vertex failures. Our approach has the interesting property that it does not rely on a compact representation of vertex cuts, and has the potential to provide optimal solutions for more vertex failures. Furthermore, we show that the DFS-based framework can be easily extended in order to answer vertex-cut queries, and the number of connected components in the presence of multiple vertex failures. In the case of three vertex failures, we can answer such queries in $O(\log n)$ time.

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arXiv2024

Connectivity Oracles for Predictable Vertex Failures

The problem of designing connectivity oracles supporting vertex failures is one of the basic data structures problems for undirected graphs. It is already well understood: previous works [Duan--Pettie STOC'10; Long--Saranurak FOCS'22] achieve query time linear in the number of failed vertices, and it is conditionally optimal as long as we require preprocessing time polynomial in the size of the graph and update time polynomial in the number of failed vertices. We revisit this problem in the paradigm of algorithms with predictions: we ask if the query time can be improved if the set of failed vertices can be predicted beforehand up to a small number of errors. More specifically, we design a data structure that, given a graph $G=(V,E)$ and a set of vertices predicted to fail $\widehat{D} \subseteq V$ of size $d=|\widehat{D}|$, preprocesses it in time $\tilde{O}(d|E|)$ and then can receive an update given as the symmetric difference between the predicted and the actual set of failed vertices $\widehat{D} \triangle D = (\widehat{D} \setminus D) \cup (D \setminus \widehat{D})$ of size $η= |\widehat{D} \triangle D|$, process it in time $\tilde{O}(η^4)$, and after that answer connectivity queries in $G \setminus D$ in time $O(η)$. Viewed from another perspective, our data structure provides an improvement over the state of the art for the \emph{fully dynamic subgraph connectivity problem} in the \emph{sensitivity setting} [Henzinger--Neumann ESA'16]. We argue that the preprocessing time and query time of our data structure are conditionally optimal under standard fine-grained complexity assumptions.

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arXiv2024

On 2-strong connectivity orientations of mixed graphs and related problems

A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is either $(u,v)$ or $(v,u)$. The problem of finding an orientation of a mixed graph that makes it strongly connected is well understood and can be solved in linear time. Here we introduce the following orientation problem in mixed graphs. Given a mixed graph $G$, we wish to compute its maximal sets of vertices $C_1,C_2,\ldots,C_k$ with the property that by removing any edge $e$ from $G$ (directed or undirected), there is an orientation $R_i$ of $G\setminus{e}$ such that all vertices in $C_i$ are strongly connected in $R_i$. We discuss properties of those sets, and we show how to solve this problem in linear time by reducing it to the computation of the $2$-edge twinless strongly connected components of a directed graph. A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. A $2$-edge twinless strongly connected component (2eTSCC) of $G$ is a maximal subset of vertices $C$ such that any two vertices $u, v \in C$ are in the same twinless strongly connected component of $G \setminus e$, for any edge $e$. These concepts are motivated by several diverse applications, such as the design of road and telecommunication networks, and the structural stability of buildings.

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arXiv2024

Online Drone Scheduling for Last-mile Delivery

Delivering a parcel from the distribution hub to the customer's doorstep is called the \textit{last-mile delivery} step in delivery logistics. In this paper, we study a hybrid {\it truck-drones} model for the last-mile delivery step, in which a truck moves on a predefined path carrying parcels and drones deliver the parcels. We define the \textsc{online drone scheduling} problem, where the truck moves in a predefined path, and the customer's requests appear online during the truck's movement. The objective is to schedule a drone associated with every request to minimize the number of drones used subject to the battery budget of the drones and compatibility of the schedules. We propose a 3-competitive deterministic algorithm using the next-fit strategy and 2.7-competitive algorithms using the first-fit strategy for the problem with $O(\log n)$ worst-case time complexity per request, where $n$ is the maximum number of active requests at any time. We also introduce \textsc{online variable-size drone scheduling} problem (OVDS). Here, we know all the customer's requests in advance; however, the drones with different battery capacities appear online. The objective is to schedule customers' requests for drones to minimize the number of drones used. We propose a $(2α+ 1)$-competitive algorithm for the OVDS problem with total running time $O(n \log n)$ for $n$ customer requests, where $α$ is the ratio of the maximum battery capacity to the minimum battery capacity of the drones. Finally, we address how to generate intervals corresponding to each customer request when there are discrete stopping points on the truck's route, from where the drone can fly and meet with the truck.

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arXiv2023

$2$-Fault-Tolerant Strong Connectivity Oracles

We study the problem of efficiently answering strong connectivity queries under two vertex failures. Given a directed graph $G$ with $n$ vertices, we provide a data structure with $O(nh)$ space and $O(h)$ query time, where $h$ is the height of a decomposition tree of $G$ into strongly connected subgraphs. This immediately implies data structures with $O(n \log{n})$ space and $O(\log{n})$ query time for graphs of constant treewidth, and $O(n^{3/2})$ space and $O(\sqrt{n})$ query time for planar graphs. For general directed graphs, we give a refined version of our data structure that achieves $O(n\sqrt{m})$ space and $O(\sqrt{m})$ query time, where $m$ is the number of edges of the graph. We also provide some simple BFS-based heuristics that seem to work remarkably well in practice. In the experimental part, we first evaluate various methods to construct a decomposition tree with small height $h$ in practice. Then we provide efficient implementations of our data structures, and evaluate their empirical performance by conducting an extensive experimental study on graphs taken from real-world applications.

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arXiv2023

Computing the $5$-Edge-Connected Components in Linear Time

We provide a deterministic algorithm for computing the $5$-edge-connected components of an undirected multigraph in linear time. There were probably good indications that this computation can be performed in linear time, but no such algorithm was actually known prior to this work. Thus, our paper answers a theoretical question, and sheds light on the possibility that a solution may exist for general $k$. A key component in our algorithm is an oracle for answering connectivity queries for pairs of vertices in the presence of at most four edge-failures. Specifically, the oracle has size $O(n)$, it can be constructed in linear time, and it answers connectivity queries in the presence of at most four edge-failures in worst-case constant time, where $n$ denotes the number of vertices of the graph. We note that this is a result of independent interest. Our paper can be considered as a follow-up of recent work on computing the $4$-edge-connected components in linear time. However, in dealing with the computation of the $5$-edge-connected components, we are faced with unique challenges that do not appear when dealing with lower connectivity. The problem is that the $4$-edge cuts in $3$-edge-connected graphs are entangled in various complicated ways, that make it difficult to organize them in a compact way. Here we provide a novel analysis of those cuts, that reveals the existence of various interesting structures. These can be exploited so that we can disentangle and collect only those cuts that are essential in computing the $5$-edge-connected components. This analysis may provide a clue for a general solution for the $k$-edge-connected components, or other related graph connectivity problems.

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arXiv2023

Connectivity Queries under Vertex Failures: Not Optimal, but Practical

We revisit once more the problem of designing an oracle for answering connectivity queries in undirected graphs in the presence of vertex failures. Specifically, given an undirected graph $G$ with $n$ vertices and $m$ edges and an integer $d_{\star}\ll n$, the goal is to preprocess the graph in order to construct a data structure $\mathcal{D}$ such that, given a set of vertices $F$ with $|F|=d\leq d_{\star}$, we can derive an oracle from $\mathcal{D}$ that can efficiently answer queries of the form ''is $x$ connected with $y$ in $G\setminus F$?''. Very recently, Long and Saranurak (FOCS 2022) provided a solution to this problem that is almost optimal with respect to the preprocessing time, the space usage, the update time, and the query time. However, their solution is highly complicated, and it seems very difficult to be implemented efficiently. Furthermore, it does not settle the complexity of the problem in the regime where $d_{\star}$ is a constant. Here, we provide a much simpler solution to this problem, that uses only textbook data structures. Our algorithm is deterministic, it has preprocessing time and space complexity $O(d_{\star}m\log n)$, update time $O(d^4 \log n)$, and query time $O(d)$. These bounds compare very well with the previous best, especially considering the simplicity of our approach. In fact, if we assume that $d_{\star}$ is a constant ($d_{\star}\geq 4$), then our algorithm provides some trade-offs that improve the state of the art in some respects. Finally, the data structure that we provide is flexible with respect to $d_{\star}$: it can be adapted to increases and decreases, in time and space that are almost proportional to the change in $d_{\star}$ and the size of the graph.

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arXiv2023

On maximal k-edge-connected subgraphs of undirected graphs

We show how to find and efficiently maintain maximal k-edge-connected subgraphs in undirected graphs. In particular, we provide the following results. (1) A general framework for maintaining the maximal k-edge-connected subgraphs upon insertions of edges or vertices, by successively partitioning the graph into its k-edge-connected components. This defines a decomposition tree, which can be maintained by using algorithms for the incremental maintenance of the k-edge-connected components as black boxes at every level of the tree. (2) As an application of this framework, we provide two algorithms for the incremental maintenance of the maximal $3$-edge-connected subgraphs. These algorithms allow for vertex and edge insertions, interspersed with queries asking whether two vertices belong to the same maximal $3$-edge-connected subgraph. The first algorithm has $O(mα(m,n) + n^2\log^2 n)$ total running time and uses $O(n)$ space, where $m$ is the number of edge insertions and queries, and $n$ is the total number of vertices inserted. The second algorithm performs the same operations in faster $O(mα(m,n) + n^2α(n,n))$ time in total, using $O(n^2)$ space. (3) We provide efficient constructions of sparse subgraphs that have the same maximal k-edge-connected subgraphs as the original graph. These are useful in speeding up computations involving the maximal k-edge-connected subgraphs in dense undirected graphs. (4) We give two deterministic algorithms for computing the maximal k-edge-connected subgraphs in undirected graphs, with running times $O(m+k^{O(1)}n\sqrt{n}\mathrm{polylog}(n))$ and $O(m+k^{O(k)}n\sqrt{n}\log{n})$, respectively. (5) A fully dynamic algorithm for maintaining information about the maximal k-edge-connected subgraphs for fixed k. Our update bounds are $O(n\sqrt{n}\log{n})$ worst-case time, and we achieve constant time for maximal k-edge-connected subgraph queries.

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arXiv2021

Computing the $4$-Edge-Connected Components of a Graph in Linear Time

We present the first linear-time algorithm that computes the $4$-edge-connected components of an undirected graph. Hence, we also obtain the first linear-time algorithm for testing $4$-edge connectivity. Our results are based on a linear-time algorithm that computes the $3$-edge cuts of a $3$-edge-connected graph $G$, and a linear-time procedure that, given the collection of all $3$-edge cuts, partitions the vertices of $G$ into the $4$-edge-connected components.

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arXiv2020

Linear-Time Algorithms for Computing Twinless Strong Articulation Points and Related Problems

A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. These concepts have several diverse applications, such as the design of telecommunication networks and the structural stability of buildings. A vertex $v \in V$ is a twinless strong articulation point of $G$ if the deletion of $v$ increases the number of TSCCs of $G$. Here, we present the first linear-time algorithm that finds all the twinless strong articulation points of a directed graph. We show that the computation of twinless strong articulation points reduces to the following problem in undirected graphs, which may be of independent interest: Given a $2$-vertex-connected (biconnected) undirected graph $H$, find all vertices $v$ that belong to a vertex-edge cut-pair, i.e., for which there exists an edge $e$ such that $H \setminus \{v,e\}$ is not connected. We develop a linear-time algorithm that not only finds all such vertices $v$, but also computes the number of edges $e$ such that $H \setminus \{v,e\}$ is not connected. This also implies that for each twinless strong articulation point $v$ which is not a strong articulation point in a strongly connected digraph $G$, we can compute the number of TSCCs in $G \setminus v$. We note that the problem of computing all vertices that belong to a vertex-edge cut-pair can be solved in linear-time by exploiting the structure of $3$-vertex-connected (triconnected) components of $H$, represented by an SPQR tree of $H$. Our approach, however, is conceptually simple, and thus likely to be more amenable to practical implementations.

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