Balanced Combinations of Solutions in Multi-Objective Optimization

For every list of integers x1, . . . , xm there is some j such that x1+ · · ·+xj −xj+1−· · ·−xm ≈ 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the xi 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.