Logarithmic hardness of the undirected edge-disjoint paths problem

We show that there is no log<sup>⅓ − ϵ</sup> <i>M</i> approximation for the undirected Edge-Disjoint Paths problem unless <i>NP</i> ⊆ <i>ZPTIME</i>(<i>n</i><sup>polylog(<i>n</i>)</sup>), where <i>M</i> is the size of the graph and ϵ is any positive constant. This hardness result also applies to the undirected All-or-Nothing Multicommodity Flow problem and the undirected Node-Disjoint Paths problem.

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