Optimizing Safe Flow Decompositions in DAGs

Network flow is one of the most studied combinatorial optimization problems having innumerable applications. Any flow on a directed acyclic graph $G$ having $n$ vertices and $m$ edges can be decomposed into a set of $O(m)$ paths. In some applications, each solution (decomposition) corresponds to some particular data that generated the original flow. Given the possibility of multiple optimal solutions, no optimization criterion ensures the identification of the correct decomposition. Hence, recently flow decomposition was studied [RECOMB22] in the Safe and Complete framework, particularly for RNA Assembly. They presented a characterization of the safe paths, resulting in an $O(mn+outR)$ time algorithm to compute all safe paths, where $outR$ is the size of the raw output reporting each safe path explicitly. They also showed that $outR$ can be $\Omega(mn^2)$ in the worst case but $O(m)$ in the best case. Hence, they further presented an algorithm to report a concise representation of the output $outC$ in $O(mn+outC)$ time, where $outC$ can be $\Omega(mn)$ in the worst case but $O(m)$ in the best case. In this work, we study how different safe paths interact, resulting in optimal output-sensitive algorithms requiring $O(m+outR)$ and $O(m+outC)$ time for computing the existing representations of the safe paths. Further, we propose a new characterization of the safe paths resulting in the {\em optimal} representation of safe paths $out_O$, which can be $\Omega(mn)$ in the worst case but requires optimal $O(1)$ space for every safe path reported, with a near-optimal computation algorithm. Overall we further develop the theory of safe and complete solutions for the flow decomposition problem, giving an optimal algorithm for the explicit representation, and a near-optimal algorithm for the optimal representation of the safe paths