Width Parameters for Minimum Flow Decomposition

Minimum flow decomposition (MFD) is the strongly NP-hard problem of finding a smallest set of integer weighted $s$-$t$ paths in an $s$-$t$ DAG $G$ whose weighted sum is equal to a given flow $f$ on $G$. Despite its many practical applications, we lack an understanding of graph structures that make MFD easy or hard. Recent progress is due to Cáceres et al. [ACM TALG 2024], who showed that the DAG width, the minimum number of paths to cover all edges, plays an essential role in the approximation of the problem. Our first set of results regard the computational complexity of MFD parameterised by the width. This question was previously open, because MFD on width-1 DAGs (paths) is trivially solvable, and the existing NP-hardness proofs use DAGs of unbounded width. We show that MFD on width-2 DAGs is already NP-hard and that MFD on width-3 DAGs is strongly NP-hard. Our main contribution complements these hardness bounds, as we show that weak NP-hardness is the best we can hope for on width-2 DAGs. In fact, we prove the more general statement that MFD with unary coded input can be solved in quasi-polynomial time on DAGs of constant parallel-width, which includes width-2 DAGs. The parallel-width of a DAG $G$ (par-width$(G)$) was defined by Deligkas and Meir [MFCS 2017] as the size of the largest minimal $s$-$t$ cut-set. We obtain these results by, a) interpreting flow decompositions as a sequence of certain digraph minor operations defined by Deligkas and Meir [MFCS 2017], and b) defining a new notion of width of a flow network, flow-width of $(G,f)$, defined as the minimum number of paths covering all edges of $G$, where every edge $e$ can be covered by at most $f(e)$ paths. Using (a) and (b), we show as an intermediate result, an improved upper bound $(\lfloor\log \Vert f\Vert\rfloor+1) \cdot \text{par-width}(G)$ for MFD, where $\Vert f\Vert$ is the largest flow weight of all edges.