726 research outputs found
Balanced Combinations of Solutions in Multi-Objective Optimization
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
On the Maximum Satisfiability of Random Formulas
Maximum satisfiability is a canonical NP-hard optimization problem that
appears empirically hard for random instances. Let us say that a Conjunctive
normal form (CNF) formula consisting of -clauses is -satisfiable if there
exists a truth assignment satisfying of all clauses
(observe that every -CNF is 0-satisfiable). Also, let denote a
random -CNF on variables formed by selecting uniformly and independently
out of all possible -clauses. It is easy to prove that for every
and every in , there is such that if , then the
probability that is -satisfiable tends to 0 as tends to
infinity. We prove that there exists a sequence such that if
then the probability that is
-satisfiable tends to 1 as tends to infinity. The sequence
tends to 0 exponentially fast in
- …