Theory and A Heuristic for the Minimum Path Flow Decomposition Problem.

Motivated by multiple genome assembly problems and other applications, we study the following minimum path flow decomposition problem: Given a directed acyclic graph $G=(V,E)$G=(V,E) with source $s$s and sink $t$t and a flow $f$f, compute a set of $s$s-$t$t paths $P$P and assign weight $w(p)$w(p) for $p\in P$p∈P such that $f(e) = \sum _{p\in P: e\in p} w(p)$f(e)=∑p∈P:e∈pw(p), $\forall e\in E$∀e∈E, and $|P|$|P| is minimized. We develop some fundamental theory for this problem, upon which we design an efficient heuristic. Specifically, we prove that the gap between the optimal number of paths and a known upper bound is determined by the nontrivial equations within the flow values. This result gives rise to the framework of our heuristic: to iteratively reduce the gap through identifying such equations. We also define an operation on certain independent substructures of the graph, and prove that this operation does not affect the optimality but can transform the graph into one with desired property that facilitates reducing the gap. We apply and test our algorithm on both simulated random instances and perfect splice graph instances, and also compare it with the existing state-of-art algorithm for flow decomposition. The results illustrate that our algorithm can achieve very high accuracy on these instances, and also that our algorithm significantly improves on the previous algorithms. An implementation of our algorithm is freely available at https://github.com/Kingsford-Group/catfish.