arXiv subjects

Francisco Sena

Publications and source records attributed to Francisco Sena.

5 recordsLinked to original sources

arXiv2026

Fast and Flexible Flow Decompositions in General Graphs via Dominators

Multi-assembly methods rely at their core on a flow decomposition problem, namely, decomposing a weighted graph into weighted paths or walks. However, most results over the past decade have focused on decompositions over directed acyclic graphs (DAGs). This limitation has lead to either purely heuristic methods, or in applications transforming a graph with cycles into a DAG via preprocessing heuristics. In this paper we show that flow decomposition problems can be solved in practice also on general graphs with cycles, via a framework that yields fast and flexible Mixed Integer Linear Programming (MILP) formulations. Our key technique relies on the graph-theoretic notion of dominator tree, which we use to find all safe sequences of edges, that are guaranteed to appear in some walk of any flow decomposition solution. We generalize previous results from DAGs to cyclic graphs, by showing that maximal safe sequences correspond to extensions of common leaves of two dominator trees, and that we can find all of them in time linear in their size. Using these, we can accelerate MILPs for any flow decomposition into walks in general graphs, by setting to (at least) 1 suitable variables encoding solution walks, and by setting to 0 other walks variables non-reachable to and from safe sequences. This reduces model size and eliminates costly linearizations of MILP variable products. We experiment with three decomposition models (Minimum Flow Decomposition, Least Absolute Errors and Minimum Path Error), on four bacterial datasets. Our pre-processing enables up to thousand-fold speedups and solves even under 30 seconds many instances otherwise timing out. We thus hope that our dominator-based MILP simplification framework, and the accompanying software library can become building blocks in multi-assembly applications.

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arXiv2026

The Power of Graph Doubling: Computing Ultrabubbles in a Bidirected Graph by Reducing to Weak Superbubbles

Bidirected graphs are a common generalisation of directed graphs where arcs can also be incoming to both their incident nodes, or outgoing from both their incident nodes. Such arcs allow a walk to change direction. Some algorithms can easily be adapted from directed graphs to bidirected graphs, such as shortest path algorithms. These adaptions are already used in practice, and implicitly use the graph doubling technique to apply an algorithm for directed graphs to bidirected graphs. In other cases, the applicability of graph doubling is not that obvious. For example, superbubbles and their generalisation to bidirected graphs ultrabubbles. Ultrabubbles are a common structure in bidirected biological graphs which carries biological meaning, but also functions as a nested clustering method, since an ultrabubble is separated by only two nodes from the rest of the graph. There is an existing method that enumerates a structure similar to ultrabubbles by enumerating (weak) superbubbles in the doubled graph. However, the literature does not make any direct connection between superbubbles and ultrabubbles except that a superbubble is an ultrabubble in a directed graph. Only a partial result connecting superbubbles and ultrabubbles exists by Harviainen et al. (2026). Graph doubling on the other hand maintains connectivity, and allows to draw a direct connection between ultrabubbles and weak superbubbles. This results in the first linear-time reduction-based algorithm for computing ultrabubbles on any bidirected graph. Together with the fact that graph doubling is already used implicitly in simple cases, our result motivates that graph doubling is a powerful yet simple technique to apply algorithms for directed graphs to bidirected graphs.

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arXiv2026

Identifying bubble-like subgraphs in linear-time via a unified SPQR-tree framework

A fundamental algorithmic problem in computational biology is to find all subgraphs of a given type (superbubbles, snarls, and ultrabubbles) in a directed or bidirected input graph. These correspond to regions of genetic variation and are useful in analyzing collections of genomes. We present the first linear-time algorithms for identifying all snarls and all ultrabubbles, resolving problems open since 2018. The algorithm for snarls relies on a new linear-size representation of all snarls with respect to the number of vertices in the graph. We employ the well-known SPQR-tree decomposition, which encodes all 2-separators of a biconnected graph. After several dynamic-programming-style traversals of this tree, we maintain key properties (such as acyclicity) that allow us to decide whether a given 2-separator defines a subgraph to be reported. A crucial ingredient for linear-time complexity is that acyclicity of linearly many subgraphs can be tested simultaneously via the problem of computing all arcs in a directed graph whose removal renders it acyclic (so-called feedback arcs). As such, we prove a fundamental result for bidirected graphs, that may be of independent interest: all feedback arcs can be computed in linear time for tipless bidirected graphs, while in general this is at least as hard as matrix multiplication, assuming the k-Clique Conjecture. Our results form a unified framework that also yields a completely different linear-time algorithm for finding all superbubbles. Although some of the results are technically involved, the underlying ideas are conceptually simple, and may extend to other bubble-like subgraphs. More broadly, our work contributes to the theoretical foundations of computational biology and advances a growing line of research that uses SPQR-tree decompositions as a general tool for designing efficient algorithms, beyond their traditional role in graph drawing.

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arXiv2025

Identifying all snarls and superbubbles in linear-time, via a unified SPQR-tree framework

Snarls and superbubbles are fundamental pangenome decompositions capturing variant sites. These bubble-like structures underpin key tasks in computational pangenomics, including structural-variant genotyping, distance indexing, haplotype sampling, and variant annotation. Snarls can be quadratically-many in the size of the graph, and since their introduction in 2018 with the vg toolkit, there has been no work on identifying all snarls in linear time. Moreover, while it is known how to find superbubbles in linear time, this result is a highly specialized solution only achieved after a long series of papers. We present the first algorithm identifying all snarls in linear time. This is based on a new representation of all snarls, of size linear in the input graph size, and which can be computed in linear time. Our algorithm is based on a unified framework that also provides a new linear-time algorithm for finding superbubbles. An observation behind our results is that all such structures are separated from the rest of the graph by two vertices (except for cases which are trivially computable), i.e. their endpoints are a 2-separator of the underlying undirected graph. Based on this, we employ the well-known SPQR tree decomposition, which encodes all 2-separators, to guide a traversal that finds the bubble-like structures efficiently. We implemented our algorithms in C++ (available at https://github.com/algbio/BubbleFinder) and evaluated them on various pangenomic datasets. Our algorithms outcompete or they are on the same level of existing methods. For snarls, we are up to two times faster than vg, while identifying all snarls. When computing superbubbles, we are up to 50 times faster than BubbleGun. Our SPQR tree framework provides a unifying perspective on bubble-like structures in pangenomics, together with a template for finding other bubble-like structures efficiently.

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arXiv2025

Safe Sequences via Dominators in DAGs for Path-Covering Problems

A path-covering problem on a directed acyclic graph (DAG) requires finding a set of source-to-sink paths that cover all the nodes, all the arcs, or subsets thereof, and additionally they are optimal with respect to some function. In this paper we study safe sequences of nodes or arcs, namely sequences that appear in some path of every path cover of a DAG. We show that safe sequences admit a simple characterization via cutnodes. Moreover, we establish a connection between maximal safe sequences and leaf-to-root paths in the source- and sink-dominator trees of the DAG, which may be of independent interest in the extensive literature on dominators. With dominator trees, safe sequences admit an O(n)-size representation and a linear-time output-sensitive enumeration algorithm running in time O(m + o), where n and m are the number of nodes and arcs, respectively, and o is the total length of the maximal safe sequences. We then apply maximal safe sequences to simplify Integer Linear Programs (ILPs) for two path-covering problems, LeastSquares and MinPathError, which are at the core of RNA transcript assembly problems from bioinformatics. On various datasets, maximal safe sequences can be computed in under 0.1 seconds per graph, on average, and ILP solvers whose search space is reduced in this manner exhibit significant speed-ups. For example on graphs with a large width, average speed-ups are in the range 50-250x for MinPathError and in the range 80-350x for LeastSquares. Optimizing ILPs using safe sequences can thus become a fast building block of practical RNA transcript assembly tools, and more generally, of path-covering problems.

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