On extensions of the deterministic online model for bipartite matching and max-sat

Abstract

The surprising results of Karp, Vazirani and Vazirani [39] and (respectively) Buchbinder et al. [18] are examples where rather simple randomization provides provably better approximations than the corresponding deterministic counterparts for online bipartite matching and (respectively) unconstrained non-monotone submodular. 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 12 approximation achieved by any greedy maximal matching algorithm and likewise cannot improve upon this approximation by any lognlog⁡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 to beat the simple deterministic greedy algorithm. For max-sat, we adapt the recent de-randomization approach of Buchbinder and Feldman [16] applied to the Buchbinder et al. [17] algorithm for max-sat to obtain a deterministic 34 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). We relate our results to previous work concerning the priority Branching Tree (pBT) model of Alekhnovich et al. [2].