Deterministic parallel algorithms for bilinear objective functions

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low independence. A series of papers, beginning with work by Luby (1988), showed that in many cases these techniques can be combined to give deterministic parallel (NC) algorithms for a variety of combinatorial optimization problems, with low time- and processor-complexity. We extend and generalize a technique of Luby for efficiently handling bilinear objective functions. One noteworthy application is an NC algorithm for maximal independent set (MIS) with $\tilde O(\log^2 n)$ time and $(m + n) n^{o(1)}$ processors; this is nearly the same as the best randomized parallel algorithms. Previous NC algorithms required either $\log^{2.5} n$ time or $mn$ processors. Other applications of our technique include algorithms of Berger (1997) for maximum acyclic subgraph and Gale-Berlekamp switching games. This bilinear factorization also gives better algorithms for problems involving discrepancy. An important application of this is to automata-fooling probability spaces, which are the basis of a notable derandomization technique of Sivakumar (2002). Previous algorithms have had very high processor complexity. We are able to greatly reduce this, with applications to set balancing and the Johnson-Lindenstrauss Lemma.