Deterministic parallel algorithms for bilinear objective functions

TL;DR

This paper introduces a generalized deterministic parallel algorithm technique for bilinear objective functions, enabling efficient derandomization and improved algorithms for combinatorial optimization, discrepancy problems, and automata-fooling in NC complexity.

Contribution

It extends Luby's technique to bilinear functions, resulting in faster NC algorithms for problems like maximal independent set and reduced processor counts for derandomization tasks.

Findings

NC algorithm for maximal independent set with near-optimal complexity

Reduced processor counts for maximum acyclic subgraph and switching games

Significant improvements in derandomization algorithms using automata-fooling

Abstract

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. On a graph $G$ with $m$ edges and $n$ vertices, this takes $\tilde O(\log^2 n)$ time and $(m + n) n^{o(1)}$ processors, nearly matching the best randomized parallel algorithms. Other applications include reduced processor counts for 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). Our method leads to large reduction in processor complexity for a number of derandomization algorithms based on automata-fooling, including set discrepancy and the Johnson-Lindenstrauss Lemma.