4 research outputs found
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. On a graph with edges and vertices, this
takes time and 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
Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel
The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability
theory, guaranteeing the existence of configurations which avoid a collection
of "bad" events which are mostly independent and have low
probability. In its simplest "symmetric" form, it asserts that whenever a
bad-event has probability and affects at most bad-events, and , then a configuration avoiding all exists.
A seminal algorithm of Moser & Tardos (2010) gives nearly-automatic
randomized algorithms for most constructions based on the LLL. However,
deterministic algorithms have lagged behind. We address three specific
shortcomings of the prior deterministic algorithms. First, our algorithm
applies to the LLL criterion of Shearer (1985); this is more powerful than
alternate LLL criteria and also removes a number of nuisance parameters and
leads to cleaner and more legible bounds. Second, we provide parallel
algorithms with much greater flexibility in the functional form of of the
bad-events. Third, we provide a derandomized version of the MT-distribution,
that is, the distribution of the variables at the termination of the MT
algorithm.
We show applications to non-repetitive vertex coloring, independent
transversals, strong coloring, and other problems. These give deterministic
algorithms which essentially match the best previous randomized sequential and
parallel algorithms.Comment: This superseded arxiv:1807.0667