Online algorithms with advice for bin packing and scheduling problems

We consider the setting of online computation with advice and study the bin packing problem and a number of scheduling problems. We show that it is possible, for any of these problems, to arbitrarily approach a competitive ratio of 1 with only a constant number of bits of advice per request. For the bin packing problem, we give an online algorithm with advice that is (1+ε)(1+ε)-competitive and uses O(1εlog⁡1ε) bits of advice per request. For scheduling on m   identical machines, with the objective function of any of makespan, machine covering and the minimization of the ℓpℓp norm, p>1p>1, we give similar results. We give online algorithms with advice which are (1+ε)(1+ε)-competitive ((1/(1−ε))(1/(1−ε))-competitive for machine covering) and also use O(1εlog⁡1ε) bits of advice per request. We complement our results by giving a lower bound that shows that for any online algorithm with advice to be optimal, for any of the above scheduling problems, a non-constant number (namely, at least (1−2mn)log⁡m, where n is the number of jobs and m is the number of machines) of bits of advice per request is needed.