On the Power of Advice and Randomization for Online Bipartite Matching
Résumé
While randomized online algorithms have access to a sequence of uniform random bits, deterministic
online algorithms with advice have access to a sequence of {\em 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 (\textsc{MBM}).
The well-known Karp-Vazirani-Vazirani algorithm \cite{kvv90} is an optimal
randomized $(1-\frac{1}{e})$-competitive algorithm for \textsc{MBM} that requires access to
$\Theta(n \log n)$ uniform random bits. We show that $\Omega(\log(\frac{1}{\epsilon}) n)$ advice
bits are necessary and $O(\frac{1}{\epsilon^5} n)$ sufficient in order to obtain a
$(1-\epsilon)$-competitive deterministic advice algorithm. Furthermore,
for a large natural class of deterministic advice algorithms, we prove that
$\Omega(\log \log \log n)$ advice bits are required in order to improve on the $\frac{1}{2}$-competitiveness of
the best deterministic online algorithm, while it is known that $O(\log n)$
bits are sufficient \cite{bkkk11}.
Last, we give a randomized online algorithm that uses $c n$ random bits, for integers $c \ge 1$,
and a competitive ratio that approaches $1-\frac{1}{e}$ very quickly as $c$ is increasing.
For example if $c = 10$, then the difference between $1-\frac{1}{e}$ and the achieved competitive ratio
is less than $0.0002$.