# A practical fpt algorithm for Flow Decomposition and transcript assembly

## TL;DR

This paper introduces a practical fixed-parameter tractable algorithm for Flow Decomposition in DAGs, crucial for transcript assembly, and demonstrates its efficiency and exactness on RNA-sequence data.

## Contribution

The paper presents the first practical linear FPT algorithm for Flow Decomposition, along with an implementation that outperforms heuristics in transcript assembly tasks.

## Key findings

- The solver finds exact solutions efficiently on RNA-sequence data.
- The algorithm's runtime is competitive with state-of-the-art heuristics.
- Hardness results show no polynomial kernels and NP-hardness for related problems.

## 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.