  For every list of integers x_1, ..., x_m there is some j such that x_1 + ...
+ x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and
for this we only need one alternation between addition and subtraction. But
what if the x_i are k-dimensional integer vectors? Using results from
topological degree theory we show that balancing is still possible, now with k
alternations.
  This result is useful in multi-objective optimization, as it allows a
polynomial-time computable balance of two alternatives with conflicting costs.
The application to two multi-objective optimization problems yields the
following results:
  - A randomized 1/2-approximation for multi-objective maximum asymmetric
traveling salesman, which improves and simplifies the best known approximation
for this problem.
  - A deterministic 1/2-approximation for multi-objective maximum weighted
satisfiability.
