(Arc-)disjoint flows in networks

4 We consider the problem of deciding whether a given network with integer capacities has two 5 feasible flows x and y with prescribed balance vectors such that the arcs that carry flow in x are 6 arc-disjoint from the arcs that carry flow in y. This generalizes a number of well-studied problems 7 such as the existence of arc-disjoint out-branchings B s , B + t where the roots s, t may be the same 8 vertex, existence of arc-disjoint spanning subdigraphs D1, D2 with prescribed degree sequences 9 in a digraph (e.g. arc-disjoint cycle factors), the weak-2-linkage problem, the number partitioning 10 problem etc. Hence the problem is NP-complete in general. We show that the problem remains 11 hard even for very restricted cases such as two arc-disjoint (s, t)-flows each of value 2 in a network 12 with capacities 1 and 2 on the arcs. On the positive side, we prove that the above problem is 13 polynomially solvable if the network is acyclic and the arc capacities as well as the desired flow 14 values are bounded. Our algorithm for this case generalizes the algorithm (by Perl and Shiloach 15 [14] for k = 2 and Fortune Hopcroft and Wyllie [11] for k ≥ 3) for the k-linkage problem in acyclic 16 digraphs. Besides, the problem is polynomial in general digraphs if all capacities are one and the 17 two flows have the same balance for all vertices in N , but remains NP-complete if the network 18 contains at least one arc with capacity 2 (and the others have capacity 1). Finally, we also show 19 that the following properties are NP-complete to decide on digraphs: the existence of a spanning 20 connected eulerian subdigraph, the existence of a cycle factor in which all cycles have even length 21 and finally the existence of a cycle factor in which all cycles have odd length. 22