DAG中安全流分解的优化
Optimizing Safe Flow Decompositions in DAGs
摘要：
网络流问题是组合优化问题中研究最多的问题之一，有着广泛的应用。具有$n$顶点和$m$边的有向无环图$G$上的任何流可以被分解成一组$O(m)$路径。在某些应用中，每个解（分解）对应于生成原始流的某些特定数据。考虑到多个最优解的可能性，没有优化标准确保正确分解的识别。因此，最近在安全和完整框架中研究了流分解[RECOMB 22]，特别是对于RNA组装。他们给出了安全路径的特征，从而产生了一个计算所有安全路径的$O(mn+out_R)$时间算法，其中$out_R$是显式报告每个安全路径的原始输出的大小。他们还表明，$out_R$在最差情况下可以是$Ω(mn^2)$，但在最佳情况下可以是$O(m)$。因此，他们进一步提出了一种算法，以在$O(mn+out_C)$时间内报告输出$out_C$的简明表示，其中$out_C$在最差情况下可以是$Ω(mn)$，但在最佳情况下可以是$O(m)$。在这项工作中，我们研究了不同的安全路径如何相互作用，从而在最佳的输出敏感的算法，需要$O(m+out_R)$和$O(m+out_C)$的时间计算现有的表示的安全路径。此外，我们提出了一个新的表征的安全路径的{ \em最佳}表示的安全路径$out_O$，它可以是$Ω(mn)$在最坏的情况下，但需要最佳的$O(1)$空间的每一个安全路径报告，与一个接近最佳的计算算法。总体上，我们进一步发展了流分解问题的安全完全解理论，给出了显式表示的最优算法和安全路径最优表示的近优算法
英文摘要：
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 $O(mn+out_R)$ time algorithm to compute all safe paths, where $out_R$ is the size of the raw output reporting each safe path explicitly. They also showed that $out_R$ can be $Ω(mn^2)$ 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 $out_C$ in $O(mn+out_C)$ time, where $out_C$ can be $Ω(mn)$ 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 $O(m+out_R)$ and $O(m+out_C)$ 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 $Ω(mn)$ 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
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