On the limitations of deterministic de-randomizations for online bipartite matching and max-sat

The surprising results of Karp, Vazirani and Vazirani and (respectively) Buchbinder et al are examples where rather simple randomizations provide provably better approximations than the corresponding deterministic counterparts for online bipartite matching and (respectively) unconstrained non-monotone submodular maximization. We show that seemingly strong extensions of the deterministic online computation model can at best match the performance of naive randomization. More specifically, for bipartite matching, we show that in the priority model (allowing very general ways to order the input stream), we cannot improve upon the trivial 1/2-approximation achieved by any greedy maximal matching algorithm and likewise cannot improve upon this approximation by any log n/log log n number of online algorithms running in parallel. The latter result yields an improved log log n - log log log n lower bound for the number of advice bits needed. For max-sat, we adapt the recent de-randomization approach of Buchbinder and Feldman applied to the Buchbinbder et al algorithm for max-sat to obtain a deterministic 3/4-approximation algorithm using width 2n parallelism. In order to improve upon this approximation, we show that exponential width parallelism of online algorithms is necessary (in a model that is more general than what is needed for the width 2n algorithm).