The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Paths problem, have been considered. Therefore, we investigate the parameterised complexity of the general Integer Multicommodity Flow problem. We show that the decision version of this problem on directed graphs for a constant number of commodities, when the capacities are given in unary, is XNLP-complete with pathwidth as parameter and XALP-complete with treewidth as parameter. When the capacities are given in binary, the problem is NP-complete even for graphs of pathwidth at most 13. We give related results for undirected graphs. These results imply that the problem is unlikely to be fixed-parameter tractable by these parameters. In contrast, we show that the problem does become fixed-parameter tractable when weighted tree partition width (a variant of tree partition width for edge weighted graphs) is used as parameter.

翻译：整数多商品流问题在文献中已被广泛研究。然而，从参数化角度来看，主要考虑的仅是特殊情况，例如不交路径问题。因此，我们研究了一般整数多商品流问题的参数化复杂性。我们证明，在有向图中，当商品数量为常数且容量以一元形式给出时，该问题的判定版本以路径宽为参数是XNLP完全的，以树宽为参数是XALP完全的。当容量以二进制形式给出时，即使对于路径宽最多为13的图，该问题也是NP完全的。我们针对无向图给出了相关结果。这些结果表明该问题不太可能通过这些参数具有固定参数可解性。相比之下，我们证明当使用加权树划分宽度（一种针对边加权图的树划分宽度变体）作为参数时，该问题确实变得固定参数可解。