On the Power of Advice and Randomization for Online Bipartite Matching

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-frac1e)-competitive algorithm for textscMBM that requires access to Theta(n log n) uniform random bits. We show that Omega(log(frac1ε) n) advice bits are necessary and O(frac1ε5 n) 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 Omega(log log log n) advice bits are required in order to improve on the frac12-competitiveness of the best deterministic online algorithm, while it is known that O(log n) bits are sufficient. Last, we give a randomized online algorithm that uses c n random bits, for integers c ≥ 1, and a competitive ratio that approaches 1-frac1e very quickly as c is increasing. For example if c = 10, then the difference between 1-frac1e and the achieved competitive ratio is less than 0.0002.