Fast and Flexible Flow Decompositions in General Graphs via Dominators

Abstract: 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 lead to either purely heuristic methods, or in applications transforming a graph with cycles into a DAG via preprocessing heuristics. In this paper we show that flow decomposition problems can be solved in practice also 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-theoretic notion of dominator tree, which we use to find all safe sequences of edges, that are guaranteed to appear in some walk of any flow decomposition solution. 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 we can find all of them in time linear in their size. Using these, we can accelerate MILPs for any flow decomposition into walks in general graphs, by setting to (at least) 1 suitable variables encoding solution walks, and by setting to 0 other walks variables non-reachable 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 pre-processing enables up to thousand-fold speedups and solves even under 30 seconds many instances otherwise timing out. We thus hope that our dominator-based MILP simplification framework, and the accompanying software library can become building blocks in multi-assembly applications.