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得到广泛研究。然而，从参数化视角来看，现有研究主要集中于特殊情形（如不交路问题）。因此，我们探究了一般整数多商品流问题的参数化复杂度。我们证明：在容量以一元表示的有向图中，对于恒定数量的商品，该问题的决策版本在以路径宽为参数时属于XNLP完全问题，在以树宽为参数时属于XALP完全问题；当容量以二进制表示时，即使路径宽不超过13的图，该问题也是NP完全的。我们还给出了无向图的相关结果。这些结果表明，该问题不可能通过上述参数实现固定参数可解。相比之下，我们证明当使用加权树分割宽（一种针对边加权图的树分割宽变体）作为参数时，该问题确实具有固定参数可解性。