  A well studied special case of bin packing is the 3-partition problem, where
n items of size > 1/4 have to be packed in a minimum number of bins of capacity
one. The famous Karmarkar-Karp algorithm transforms a fractional solution of a
suitable LP relaxation for this problem into an integral solution that requires
at most O(log n) additional bins.
  The three-permutations-problem of Beck is the following. Given any 3
permutations on n symbols, color the symbols red and blue, such that in any
interval of any of those permutations, the number of red and blue symbols is
roughly the same. The necessary difference is called the discrepancy.
  We establish a surprising connection between bin packing and Beck's problem:
The additive integrality gap of the 3-partition linear programming relaxation
can be bounded by the discrepancy of 3 permutations.
  Reversely, making use of a recent example of 3 permutations, for which a
discrepancy of Omega(log n) is necessary, we prove the following: The O(log^2
n) upper bound on the additive gap for bin packing with arbitrary item sizes
cannot be improved by any technique that is based on rounding up items. This
lower bound holds for a large class of algorithms including the Karmarkar-Karp
procedure.
