  We show that in a knapsack feasibility problem an integral vector $p$, which
is short, and near parallel to the constraint vector gives a branching
direction with small integer width.
  We use this result to analyze two computationally efficient reformulation
techniques on low density knapsack problems. Both reformulations have a
constraint matrix with columns reduced in the sense of Lenstra, Lenstra, and
Lov\'asz. We prove an upper bound on the integer width along the last variable,
which becomes 1, when the density is sufficiently small.
  In the proof we extract from the transformation matrices a vector which is
near parallel to the constraint vector $a.$ The near parallel vector is a good
branching direction in the original knapsack problem, and this transfers to the
last variable in the reformulations.
