DIMACS TR: 96-37
Superpolynomial Lower Bounds for Monotone Span Programs
Authors: Laszlo Babai, Anna Gal, Avi Wigderson
ABSTRACT
In this paper we obtain the first superpolynomial lower bounds for {\it
monotone span programs} computing explicit functions. The best previous
lower bound was $\Omega(n^{5/2})$ by Beimel, G\'al, Paterson \cite{BGP};
our proof exploits a general combinatorial lower bound criterion from
that paper. Our lower bounds are based on an analysis of Paley-type
bipartite graphs via Weil's character sum estimates. We prove an
$n^{\Omega ( \log n / \log\log n)}$ lower bound for an explicit family
of monotone Boolean functions in $n$ variables, which implies the same
lower bound for the size of monotone span programs for the clique problem.
Our results give the first superpolynomial lower bounds for linear secret
sharing schemes.
We demonstrate the surprising power of monotone span programs by exhibiting
a function computable in this model in linear size while requiring
superpolynomial size monotone circuits and exponential size monotone
formulae. We also show that the perfect matching function can be computed
by polynomial size (non-monotone) span programs over arbitrary fields.
Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1996/96-37.ps.gz
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