Cut arcs , or strong bridges , are one of the most fundamental reachability notions in directed graphs. Specifically, in a strongly connected graph \(G=(V,E)\) ( \(|V|=n\) , \(|E|=m\) ), a cut arc is an arc \(e\in E\) for which there exist \(u,v\in V\) , such that all \( u \) - \( v \) walks contain \( e \) . In this article, we generalise this notion to cut paths , that is, walks \( W \) for which there exist \(u,v\in V\) , such that all \( u \) - \( v \) walks contain \( W \) as subwalk. We first prove various properties of cut paths and define their remainder structure , which we use to present a simple \(O(m)\) -time verification algorithm for a cut path. We further show that a graph contains at most \(O(n)\) maximal cut paths of length at most \(O(n)\) each, and present an optimal \(O(n^{2})\) enumeration algorithm for maximal cut paths. We apply cut paths and their remainder structure to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. Cut paths provide simple \(O(m)\) -time algorithms verifying if a walk is safe or multi-safe. Further, by simultaneous computation of remainder structures of all subwalks of a cut path in linear time, we can identify all maximal multi-safe walks in \(O(mn)\) time. This improves over the state-of-the-art algorithm running in time \(O(m^{2}+n^{3}\log n)\) .

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation X on a directed graph G into weighted source-to-sink paths whose weighted sum equals X . We show that, for acyclic graphs, considering the width of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of width-stable graphs, for which a popular heuristic is a O (log Val ( X ))-approximation ( Val ( X ) being the total flow of X ), and strengthen its worst-case approximation ratio from \(\Omega (\sqrt {m})\) to Ω ( m /log m ) for sparse graphs, where m is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a (⌈ log ‖ X ‖ ⌉ +1)-approximation (‖ X ‖ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations (‖ X ‖ ≤ 1), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ 2018 ], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

In a strongly connected graph $G = (V,E)$, a cut arc (also called strong bridge) is an arc $e \in E$ whose removal makes the graph no longer strongly connected. Equivalently, there exist $u,v \in V$, such that all $u$-$v$ walks contain $e$. Cut arcs are a fundamental graph-theoretic notion, with countless applications, especially in reachability problems. In this paper we initiate the study of cut paths, as a generalisation of cut arcs, which we naturally define as those paths $P$ for which there exist $u,v \in V$, such that all $u$-$v$ walks contain $P$ as subwalk. We first prove various properties of cut paths and define their remainder structures, which we use to present a simple $O(m)$-time verification algorithm for a cut path ($|V| = n$, $|E| = m$). Secondly, we apply cut paths and their remainder structures to improve several reachability problems from bioinformatics. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. We show that cut paths provide simple $O(m)$-time algorithms verifying if a walk is safe or multi-safe. For multi-safety, we present the first linear time algorithm, while for safety, we present a simple algorithm where the state-of-the-art employed complex data structures. Finally we show that the simultaneous computation of remainder structures of all subwalks of a cut path can be performed in linear time. These properties yield an $O(mn)$ algorithm outputting all maximal multi-safe walks, improving over the state-of-the-art algorithm running in time $O(m^2+n^3)$. The results of this paper only scratch the surface in the study of cut paths, and we believe a rich structure of a graph can be revealed, considering the perspective of a path, instead of just an arc.

Network flow is one of the most studied combinatorial optimization problems having innumerable applications. Any flow on a directed acyclic graph G having n vertices and m edges can be decomposed into a set of O(m) paths. The applications of such a flow decomposition range from network routing to the assembly of biological sequences. However, in some applications, each solution (decomposition) corresponds to some particular data that generated the original flow. Given the possibility of multiple optimal solutions, no optimization criterion ensures the identification of the correct decomposition. Hence, recently flow decomposition was studied [RECOMB22] in the Safe and Complete framework, particularly for RNA Assembly. The proposed solution reported all the safe paths, i.e., the paths which are subpath of every possible solution of flow decomposition. They presented a characterization of the safe paths, resulting in an O(mn+out_R) time algorithm to compute all safe paths, where out_R is the size of the raw output reporting each safe path explicitly. They also showed that out_R can be Ω(mn²) in the worst case but O(m) in the best case. Hence, they further presented an algorithm to report a concise representation of the output out_C in O(mn+out_C) time, where out_C can be Ω(mn) in the worst case but O(m) in the best case. In this work, we study how different safe paths interact, resulting in optimal output-sensitive algorithms requiring O(m+out_R) and O(m+out_C) time for computing the existing representations of the safe paths. Our algorithm uses a novel data structure called Path Tries, which may be of independent interest. Further, we propose a new characterization of the safe paths resulting in the optimal representation of safe paths out_O, which can be Ω(mn) in the worst case but requires optimal O(1) space for every safe path reported. We also present a near-optimal algorithm to compute all the safe paths in O(m+out_Olog n) time. The new representation also establishes tighter worst case bounds Θ(mn²) and Θ(mn) bounds for out_R and out_C (along with out_O), respectively. Overall we further develop the theory of safe and complete solutions for the flow decomposition problem, giving an optimal algorithm for the explicit representation, and a near-optimal algorithm for the optimal representation of the safe paths.

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation $X$ on a directed graph $G$ into weighted source-to-sink paths whose superposition equals $X$. We show that, for acyclic graphs, considering the \emph{width} of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of \emph{width-stable} graphs, for which a popular heuristic is a \gwsimple-approximation ($|X|$ being the total flow of $X$), and strengthen its worst-case approximation ratio from $Ω(\sqrt{m})$ to $Ω(m / \log m)$ for sparse graphs, where $m$ is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a $(\lceil \log \Vert X \Vert \rceil +1)$-approximation ($\Vert X \Vert$ being the maximum absolute value of $X$ on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations ($\Vert X \Vert \leq 1$), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

Abstract Given a directed graphGand a pair of nodessandt, ans-tbridgeofGis an edge whose removal breaks alls-tpaths ofG(and thus appears in alls-tpaths). Computing alls-tbridges ofGis a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of “safety” from bioinformatics. We say that a walkWissafewith respect to a set $${\mathcal {W}}$$ W ofs-twalks, ifWis a subwalk of all walks in $${\mathcal {W}}$$ W . We start by considering the maximal safe walks when $${\mathcal {W}}$$ W consists of: alls-tpaths, alls-ttrails, or alls-twalks ofG. We show that the solutions for the first two problems immediately follow from finding alls-tbridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset ofvisibleedges. Here we prove a dichotomy between thes-tpaths ands-ttrails cases, and thes-twalks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations ofs-tarticulation points(nodes appearing in alls-tpaths). We thus obtain the best possible results for natural “safety”-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.

Genome assembly is a fundamental problem in Bioinformatics, where for a given set of overlapping substrings of a genome, the aim is to reconstruct the source genome. The classical approaches to solving this problem use assembly graphs, such as de Bruijn graphs or overlap graphs, which maintain partial information about such overlaps. For genome assembly algorithms, these graphs present a trade-off between overlap information stored and scalability. Thus, Hierarchical Overlap Graph (HOG) was proposed to overcome the limitations of both these approaches. For a given set $P$ of $n$ strings, the first algorithm to compute HOG was given by Cazaux and Rivals [IPL20] requiring $O(||P||+n^2)$ time using superlinear space, where $||P||$ is the cumulative sum of the lengths of strings in $P$. This was improved by Park et al. [SPIRE20] to $O(||P||\log n)$ time and $O(||P||)$ space using segment trees, and further to $O(||P||\frac{\log n}{\log \log n})$ for the word RAM model. Both these results described an open problem to compute HOG in optimal $O(||P||)$ time and space. In this paper, we achieve the desired optimal bounds by presenting a simple algorithm that does not use any complex data structures. At its core, our solution improves the classical result [IPL92] for a special case of the All Pairs Suffix Prefix (APSP) problem from $O(||P||+n^2)$ time to optimal $O(||P||)$ time, which may be of independent interest.

Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G. Similarly, an s-t articulation point of G is a node whose removal breaks all s-t paths of G. Computing the sequence of all s-t bridges of G (as well as the s-t articulation points) is a basic graph problem, solvable in linear time using the classical min-cut algorithm (Ford and Fulkerson, 1956). We show a simplified and self-contained algorithm computing all s-t bridges and s-t articulation points of G, based on a single graph traversal from s to t avoiding an arbitrary s-t path, which is interrupted at the s-t bridges. Its proof of correctness uses simple inductive arguments, making the problem an application of merely graph traversal, rather than of the more complex maximum flow problem.

In recent years, significant advances have been made in the design and analysis of fully dynamic maximal matching algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. In this paper, we attempt to bridge the gap between theory and practice that is currently observed for the fully dynamic maximal matching problem. We engineer several algorithms and empirically study those algorithms on an extensive set of dynamic instances.

Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical DFS algorithm requires $O(m+n)$ time for a graph having $n$ vertices and $m$ edges. In the streaming model, an algorithm is allowed several passes (preferably single) over the input graph having a restriction on the size of local space used. Trivially, a DFS tree can be computed using a single pass using $O(m)$ space. In the semi-streaming model allowing $O(n)$ space, it can be computed in $O(n)$ passes, where each pass adds one vertex to the DFS tree. However, it remains an open problem to compute a DFS tree using $o(n)$ passes using $o(m)$ space even in any relaxed streaming environment. We present the first semi-streaming algorithms that compute a DFS tree of an undirected graph in $o(n)$ passes using $o(m)$ space. We first describe an extremely simple algorithm that requires at most $\lceil n/k\rceil$ passes using $O(nk)$ space, where $k$ is any positive integer. We then improve this algorithm by using more involved techniques to reduce the number of passes to $\lceil h/k\rceil$ under similar space constraints, where $h$ is the height of the computed DFS tree. In particular, this algorithm improves the bounds for the case where the computed DFS tree is shallow (having $o(n)$ height). Moreover, this algorithm is presented as a framework that allows the flexibility of using any algorithm to maintain a DFS tree of a stored sparser subgraph as a black box, which may be of independent interest. Both these algorithms essentially demonstrate the existence of a trade-off between the space and number of passes required for computing a DFS tree. Furthermore, we evaluate these algorithms experimentally which reveals their exceptional performance in practice. For both random and real graphs, they require merely a few passes even when allowed just $O(n)$ space.

Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes $O(m+n)$ time to build a DFS tree for a given undirected graph $G=(V,E)$ on $n$ vertices and $m$ edges. We address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem: (1) Fault tolerant DFS tree: There exists a data structure of size $\tilde{O}(m)$ (where $\tilde{O}()$ hides the polylogarithmic factors) which can be preprocessed in $\tilde{O}(m)$ time such that given any set ${\cal F}$ of failed vertices or edges, a DFS tree of the graph $G\setminus {\cal F}$ can be reported in $\tilde{O}(n|{\cal F}|)$ time. (2) Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes $\tilde{O}(m)$ time for preprocessing and worst case $\tilde{O}(\sqrt{mn})$ time per update for any arbitrary online sequence of updates. (3) Incremental DFS tree: There exists an incremental algorithm for maintaining a DFS tree that takes $\tilde{O}(m)$ time for preprocessing and worst case $\tilde{O}(n)$ time per update for any arbitrary online sequence of edge insertion. These are the first $o(m)$ worst case time results for maintaining a DFS tree of a dense graph in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm for dense graphs with $O(1)$ query time and $o(m)$ worst case update time for connectivity, biconnectivity, and 2-edge connectivity in the dynamic subgraph model.

The depth first search (DFS) tree is a fundamental data structure used for solving various graph problems. For a given graph G = (V, E) on n vertices and m edges, a DFS tree can be built in O(m + n) time. In the last 20 years, a few algorithms have been designed for maintaining a DFS tree efficiently under insertion of edges. For undirected graphs, there are two prominent algorithms, namely, ADFS1 and ADFS2 [ICALP14] that achieve total update time of and O(n2) respectively. For directed acyclic graphs, the only non-trivial algorithm, namely, FDFS [IPL97] requires total O(mn) update time. However, even after 20 years of this result, there does not exist any non-trivial incremental algorithm for maintaining a DFS tree in directed graphs with o(m2) worst case bound. In this paper, we carry out extensive experimental and theoretical evaluation of the existing incremental DFS algorithms in random graphs and real world graphs and derive the following results. 1. For insertion of a uniformly random sequence of edges, each of ADFS1, ADFS2 and FDFS perform equally well and are found to take Θ(n2) time experimentally. This is quite surprising because the worst case bounds of ADFS1 and FDFS are greater than Θ(n2) by a factor of and m/n respectively, which are also proven to be tight. We complement this experimental result with a probabilistic analysis of these algorithms establishing Õ(n2) bound on their time complexity. For this purpose, we derive results about the structure of a DFS tree in a random graph. These results are of independent interest in the domain of random graphs. 2. The insight that we developed about DFS tree in random graphs leads us to design an extremely simple algorithm for incremental DFS that works for both undirected and directed graphs. Moreover, this algorithm theoretically matches and experimentally outperforms the state-of-the-art algorithm in dense random graphs. Furthermore, it can also be used as a single-pass semi-streaming algorithm for computing incremental DFS and strong connectivity for random graphs using O(n log n) space. 3. Even for real world graphs, which are usually sparse, both ADFS1 and FDFS turn out to be much better than their theoretical bounds. Here again, we present two simple algorithms for incremental DFS for directed and undirected graphs respectively, which perform very well on real graphs. In fact our proposed algorithm for directed graphs almost always matches the performance of FDFS.

Let G=(V,E) be a graph with n vertices and m edges, with a designated set of sigma sources S subseteq V. The fault tolerant subgraph for any graph problem maintains a sparse subgraph H=(V,E') of G with E' subseteq E, such that for any set F of k failures, the solution for the graph problem on G\F is maintained in its subgraph H\F. We address the problem of maintaining a fault tolerant subgraph for computing Breath First Search tree (BFS) of the graph from a single source s in V (referred as k FT-BFS) or multiple sources S subseteq V (referred as k FT-MBFS). We simply refer to them as FT-BFS (or FT-MBFS) for k=1, and dual FT-BFS (or dual FT-MBFS) for k=2. The problem of k FT-BFS was first studied by Parter and Peleg [ESA13]. They designed an algorithm to compute FT-BFS subgraph of size O(n^{3/2}). Further, they showed how their algorithm can be easily extended to FT-MBFS requiring O(sigma^{1/2}n^{3/2}) space. They also presented matching lower bounds for these results. The result was later extended to solve dual FT-BFS by Parter [PODC15] requiring (n^{5/3}) space, again with matching lower bounds. However, their result was limited to only edge failures in undirected graphs and involved very complex analysis. Moreover, their solution doesn't seems to be directly extendible for dual FT-MBFS problem. We present a similar algorithm to solve dual FT-BFS problem with a much simpler analysis. Moreover, our algorithm also works for vertex failures and directed graphs, and can be easily extended to handle dual FT-MBFS problem, matching the lower bound of O(sigma^{1/3}n^{5/3}) space described by Parter [PODC15]. The key difference in our approach is a much simpler classification of path interactions which formed the basis of the analysis by Parter [PODC15].

Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical algorithm [SIAMCOMP74] for building a DFS tree requires O(m+n) time for a given undirected graph G having n vertices and m edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating the DFS tree of an undirected graph after an edge/vertex update in O (n) time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel environment, a DFS tree can be computed from scratch using O(m) processors in expected O (1) time [SICOMP90] on an EREW PRAM, whereas the best deterministic algorithm takes O (√n) time [SIAMCOMP90,JAL93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors) deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph.

Given an undirected graph G = (V, E) on n vertices and m edges, we address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem. 1. Fault tolerant DFS tree: There exists a data structure of size Õ(m)1 such that given any set ℱ of failed vertices or edges, a DFS tree of the graph G\ℱ can be reported in Õ(n|ℱ|) time. 2. Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst case time per update for any arbitrary online sequence of updates. 3. Incremental DFS tree: Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree in Õ(n) worst case time per edge insertion. These are the first o(m) worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with O(1) query time and o(m) worst case update time for the dynamic subgraph connectivity, biconnectivity, and 2-edge connectivity.