A practical fpt algorithm for Flow Decomposition and transcript assembly

Abstract

The Flow Decomposition problem, which asks for the smallest set of weighted paths that "covers" a flow on a DAG, has recently been used as an important computational step in transcript assembly. We prove the problem is in FPT when parameterized by the number of paths by giving a practical linear fpt algorithm. Further, we implement and engineer a Flow Decomposition solver based on this algorithm, and evaluate its performance on RNA-sequence data. Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic. Finally, we contextualize our design choices with two hardness results related to preprocessing and weight recovery. Specifically, k-Flow Decomposition does not admit polynomial kernels under standard complexity assumptions, and the related problem of assigning (known) weights to a given set of paths is NP-hard.

Strain-aware assembly of genomes from mixed samples using flow variation graphs

“…Many approximation algorithms have been developed for finding a minimum path flow decomposition, e.g. [40,41], but these algorithms could not even handle our smallest data set (a mixture of 2 haplotypes of length 2500 bp). Therefore, we resort to other, more efficient means for obtaining a set of haplotypes from the given flow solution [42].…”

Flow Decomposition With Subpath Constraints

“…Based on this reduction, in Section 3.2 we also propose a "bridge-reweighting" heuristic algorithm to solve the minimum FDSC problem. Additionally, we give an FPT algorithm for the minimum FDSC problem (Theorem 20), extending the one of Kloster et al [9]. Finally, in Section 4, to add to the complexity picture around the FDSC problem, we show that two application-oriented FD problems related to FDSC are NP-hard in the strong sense, even without requiring a solution with a minimum number of paths.…”

“…In Kloster et al [9] it was shown that in the case of RNA transcripts, most of the time the "true" transcripts also provide a minimum flow decomposition of the splice graph. However, there can often be more than one solution to the minimum flow decomposition problem; indeed, Kloster et al found that, when the number of true transcripts is seven, the minimum flow decomposition found corresponds to the true paths in only 80% of the instances of that size, with lower accuracies as the number of true paths increases.…”

“…Finding a flow decomposition with the minimum number of weighted paths is a well-studied problem in computer science. Even when restricted to DAGs, the minimum FD problem is NP-hard [11], and thus various practical approaches to it exist: approximation algorithms [12], [13], [14], [15], [16], [17], FPT algorithms [9], greedy algorithms [11], [18]. By taking the set of subpaths constraints to be empty (or to correspond to all edges of the graph with non-zero flow), it follows that also finding a solution to the FDSC problem with a minimum number of paths is NP-hard.…”

“…We experiment on a dataset developed by Shao et al [18] to study their heuristic for the minimum FD problem. The same dataset was then used by Kloster et al [9], who focused on studying the usefulness of standard minimum flow decompositions in RNA assembly, as explained above. We find that our FDSC algorithms increase our ability to uncover the ground truth RNA transcripts, as more and longer subpath constraints are included in the input.…”