The persistent homology transform, Betti function transform, and Euler characteristic transform represent a shape with a multiset of persistence diagrams, Betti functions, or Euler characteristic functions, respectively, parameterized by the sphere of directions in the ambient space. In this work, we give the first explicit construction of finite sets of directions discretizing the verbose variants of these transforms and show that such discretizations faithfully represent the underlying shape. Our discretization, while exponential in the dimension of the shape, does not depend on any restrictions on the particular immersion beyond general position, and is stable with respect to various perturbations.

Abstract Minimum flow decomposition (MFD) — the problem of finding a minimum set of weighted source-to-sink paths that perfectly decomposes a flow — is a classical problem in Computer Science, and variants of it are powerful models in a different fields such as Bioinformatics and Transportation. Even on acyclic graphs, the problem is NP-hard, and most practical solutions have been via heuristics or approximations. While there is an extensive body of research on acyclic graphs, currently there is no exact solution on graphs with cycles. In this paper we present the first ILP formulation for three natural variants of the MFD problem in graphs with cycles, asking for a decomposition consisting only of weighted source-to-sink paths or cycles, trails, and walks, respectively. On three datasets of increasing levels of complexity from both Bioinformatics and Transportation, our approaches solve any instance in under 12 minutes. Our implementations are freely available at https://github.com/algbio/MFD-ILP .

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.

MOTIVATION: Many important problems in Bioinformatics (e.g. assembly or multiassembly) admit multiple solutions, while the final objective is to report only one. A common approach to deal with this uncertainty is finding "safe" partial solutions (e.g. contigs) which are common to all solutions. Previous research on safety has focused on polynomially time solvable problems, whereas many successful and natural models are NP-hard to solve, leaving a lack of "safety tools" for such problems. We propose the first method for computing all safe solutions for an NP-hard problem, "minimum flow decomposition" (MFD). We obtain our results by developing a "safety test" for paths based on a general integer linear programming (ILP) formulation. Moreover, we provide implementations with practical optimizations aimed to reduce the total ILP time, the most efficient of these being based on a recursive group-testing procedure. RESULTS: Experimental results on transcriptome datasets show that all safe paths for MFDs correctly recover up to 90% of the full RNA transcripts, which is at least 25% more than previously known safe paths. Moreover, despite the NP-hardness of the problem, we can report all safe paths for 99.8% of the over 27 000 non-trivial graphs of this dataset in only 1.5 h. Our results suggest that, on perfect data, there is less ambiguity than thought in the notoriously hard RNA assembly problem. AVAILABILITY AND IMPLEMENTATION: https://github.com/algbio/mfd-safety.

Flow network decomposition is a natural model for problems where we are given a flow network arising from superimposing a set of weighted paths and would like to recover the underlying data, i.e., decompose the flow into the original paths and their weights. Thus, variations on flow decomposition are often used as subroutines in multiassembly problems such as RNA transcript assembly. In practice, we frequently have access to information beyond flow values in the form of subpaths, and many tools incorporate these heuristically. But despite acknowledging their utility in practice, previous work has not formally addressed the effect of subpath constraints on the accuracy of flow network decomposition approaches. We formalize the flow decomposition with subpath constraints problem, give the first algorithms for it, and study its usefulness for recovering ground truth decompositions. For finding a minimum decomposition, we propose both a heuristic and an FPT algorithm. Experiments on RNA transcript datasets show that for instances with larger solution path sets, the addition of subpath constraints finds 13% more ground truth solutions when minimal decompositions are found exactly, and 30% more ground truth solutions when minimal decompositions are found heuristically.

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.

Flow network decomposition is a natural model for problems where we are given a flow network arising from superimposing a set of weighted paths and would like to recover the underlying data, i.e., decompose the flow into the original paths and their weights. Thus, variations on flow decomposition are often used as subroutines in multiassembly problems such as RNA transcript assembly. In practice, we frequently have access to information beyond flow values in the form of subpaths, and many tools incorporate these heuristically. But despite acknowledging their utility in practice, previous work has not formally addressed the effect of subpath constraints on the accuracy of flow network decomposition approaches. We formalize the flow decomposition with subpath constraints problem, give the first algorithms for it, and study its usefulness for recovering ground truth decompositions. For finding a minimum decomposition, we propose both a heuristic and an FPT algorithm. Experiments on RNA transcript datasets show that for instances with larger solution path sets, the addition of subpath constraints finds 13% more ground truth solutions when minimal decompositions are found exactly, and 30% more ground truth solutions when minimal decompositions are found heuristically.

RNA-Seq technology allows for high-throughput, low cost measurement of gene expression. An important step in this process is the assembly of mRNA transcript short reads into full transcripts. The problem can be viewed as a flow decomposition problem in which the objective is to minimize the number of path flows needed to represent a given flow. In this work we relax the edge flow constraints to allow for some uncertainty in their measurement. We formulate this as the Inexact Flow Decomposition problem and propose an algorithmic strategy to solve it. In practice, real biological data has measurement errors and so experimentally-derived edge-weighted splice graphs are often not flows. The proposed method is the first approach to this problem that explicitly controls the error allowed on each edge in these graphs in order to achieve a flow. In an intermediate step, the method solves an exact flow decomposition instance; if a greedy method is used for this step, the overall running time is O(|E|2|V|2+|P|3), where P is the solution found to the flow decomposition instance. Preliminary results on simulated biological data sets show that in many cases the ground truth paths can be recovered at approximately correct abundances, even with noisy input data.