Given an undirected graph $G=(V,E)$, two distinct vertices $s,t\in V$, a weight function $f:E \to \mathbb{N}$, and a constant $M\in \mathbb{N}$, does there exist a pair of edge disjoint paths connecting $s$ and $t$, each of which of weight at most $M$?
Call this problem DPBP (for disjoint pair of bounded paths).
Finding a pair of edge disjoint paths with bounded sum of weights is in $POLY$ [Suurballe's algorithm] http://en.wikipedia.org/wiki/Suurballe%27s_algorithm)).
Is DPBP NP-complete?