Width Helps and Hinders Splitting Flows

We show that, for acyclic graphs, considering the width of the graph yields advances in our understanding of MFD approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of width-stable graphs, for which a popular heuristic is a $O(\log Val(X))$-approximation ($Val(X)$ being the total flow of $X$), and strengthen its worst-case approximation ratio from $\Omega(\sqrt{m})$ to $\Omega(m/\log{m})$ for sparse graphs. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a $(\lceil \log ||X|| \rceil + 1)$-approximation ($||X||$ being the maximum absolute value of $X$ on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations ($||X||\le 1$), using a generalised notion of width for this problem.