Two-Way Rounding

x n ∈ {0, 1} so that the partial sums ¯ x 1 + · · · + ¯ x k and ¯ x σ1 + · · · + ¯ x σk differ from the unrounded values x 1 + · · · + x k and x σ1 + · · · + x σk by at most n/(n + 1), for 1 ≤ k ≤ n. The latter bound is best possible. The proof uses an elementary argument about flows in a certain network, and leads to a simple algorithm that finds an optimum way to round. Many combinatorial optimization problems in integers can be solved or approximately solved by first obtaining a real-valued solution and then rounding to integer values. Spencer [11] proved that it is always possible to do the rounding so that partial sums in two independent orderings are properly rounded. His proof was indirect—a corollary of more general results [7] about discrepancies of set systems—and it guaranteed only that the rounded partial sums would differ by at most 1 − 2 −2 n from the unrounded values. The purpose of this note is to give a more direct proof, which leads to a sharper result.