While randomized online algorithms have access to a sequence of uniform
random bits, deterministic online algorithms with advice have access to a
sequence of advice bits, i.e., bits that are set by an all powerful oracle
prior to the processing of the request sequence. Advice bits are at least as
helpful as random bits, but how helpful are they? In this work, we investigate
the power of advice bits and random bits for online maximum bipartite matching
(MBM).
The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized
(1−e1)-competitive algorithm for \textsc{MBM} that requires access
to Θ(nlogn) uniform random bits. We show that
Ω(log(ϵ1)n) advice bits are necessary and
O(ϵ51n) sufficient in order to obtain a
(1−ϵ)-competitive deterministic advice algorithm. Furthermore, for a
large natural class of deterministic advice algorithms, we prove that
Ω(logloglogn) advice bits are required in order to improve on the
21-competitiveness of the best deterministic online algorithm, while
it is known that O(logn) bits are sufficient.
Last, we give a randomized online algorithm that uses cn random bits, for
integers c≥1, and a competitive ratio that approaches 1−e1
very quickly as c is increasing. For example if c=10, then the difference
between 1−e1 and the achieved competitive ratio is less than
0.0002