Is Multicommodity Flow problem really NP-hard?

Within paper ON THE COMPLEXITY OF TIMETABLE AND MULTI-COMMODITY FLOW PROBLEMS

there's a proof that the multi-commodity is NP-hard.

While in Multicommodity flows over time: Efficient algorithms and complexity

authors states that "[...] the only known polynomial-time algorithms for static multicommodity flow computations require general linear programming techniques".

They use word "static" only to differ from its time-dependent version. But I think it is the same problem considered in the first paper. Authors also refer to its complexity with the wording "poly($$\simeq$$LP)".

Can someone help me understand the right way of reading this apparent contradiction?

1 Answer

"ON THE COMPLEXITY OF TIMETABLE AND MULTI-COMMODITY FLOW PROBLEMS" talks about integral flow while "Multicommodity flows over time: Efficient algorithms and complexity" does not. The integrality constraint (flows being integer valued) is important see LP vs ILP.