Fast and Flexible Flow Decompositions in General Graphs via Dominators

Multi-assembly methods rely at their core on a flow decomposition problem, namely, decomposing a weighted graph into weighted paths or walks. However, most results over the past decade have focused on decompositions over directed acyclic graphs (DAGs). This limitation has led to either purely heuristic methods or, in applications, transforming a graph with cycles into a DAG via preprocessing heuristics. In this article, we show that flow decomposition problems can also be solved in practice on general graphs with cycles via a framework that yields fast and flexible mixed-integer linear programming (MILP) formulations. Our key technique relies on the graph-theoretical notion of a dominator tree, which we use to find all safe sequences of edges that are guaranteed to appear in some walk of any flow decomposition. We generalize previous results from DAGs to cyclic graphs by showing that maximal safe sequences correspond to extensions of common leaves of two dominator trees, and that all such sequences can be found in time linear in their size. Using these, we can accelerate MILPs for any flow decomposition into walks in general graphs by setting suitable variables encoding solution walks to (at least) 1 and by setting to 0 other walk variables that are nonreachable to and from safe sequences. This reduces model size and eliminates costly linearizations of MILP variable products. We experiment with three decomposition models (minimum flow decomposition, least absolute errors, and minimum path error) on four bacterial datasets. Our preprocessing enables up to 1000-fold speedups and solves many instances that would otherwise time out in under 30 seconds. We thus hope that our dominator-based MILP simplification framework, together with the accompanying software library, can serve as building blocks for multi-assembly applications.